Is parametric Bayesian inference a special case of nonparametric Bayesian inference? I'm thinking about univariate density estimation.
Original Question
In parametric inference, you assume the data are generated from a density that can be summarized by finitely-many parameters. You place a prior on those parameters and try to make inference about them.
In nonparametric inference, you don't make a parametric assumption. You place a prior on the (infinite-dimensional) space of probability density functions and try to make inference about the density function directly.
Question: is parametric inference just a special case of nonparametric inference? That is, by making a parametric assumption and placing a prior on the parameters, are you simply defining a very dogmatic prior on the unrestricted space of densities - one that places no "mass" on densities outside the parametric family?
Addendum
I think the following is closer to what I had in mind when I asked the original question. Denote the space of probability density functions by 
$$\mathscr{D}=\left\{f:\mathbb{R}\to[0,\,\infty):\int f(x)dx=1\right\},$$
and denote a generic parametric family by 
$$\mathscr{P}=\{f_\theta\in\mathscr{D}:\theta\in\Theta\subseteq\mathbb{R}^p\}.$$
If you place a probability measure $P$ on $(\Theta,\,\mathcal{B})$, I think it induces a probability measure on the infinite-dimensional function space $\mathscr{P}$ as follows: 


*

*For each Borel set $E\in\mathcal{B}$, define 
$$F(E):=\{f_\theta\in\mathscr{P}:\theta\in E\};$$

*You can show that $\mathcal{F}(\mathcal{B}):=\{F(E):E\in\mathcal{B}\}$ is a $\sigma$-algebra on $\mathscr{P}$; 

*Set $P(f_\theta\in F(E)):=P(\theta\in E)$ for each $F(E)\in\mathcal{F}(\mathcal{B})$.
Question: In what sense does the probability measure $P$ on $(\Theta,\,\mathcal{B})$ induce a probability measure on all of $\mathscr{D}$?
 A: The answer to your question is "yes" within the context of the particular nonparametric framework that I describe below. 
Bayesian density "estimation" can understood as computing a predictive distribution for a "future" observation $y_{n+1}$ based on a collection of observations $y_{1:n} = (y_1,\ldots,y_n)$. Suppose the observations are conditionally independent:
\begin{equation}
p(y_{1:n+1}|\psi) = \prod_{i=1}^{n+1} p(y_i|\psi) 
\end{equation}
for some $p(y_i|\psi)$ where $\psi$ is a parameter (vector). 
The predictive distribution can be expressed as follows:
\begin{equation}\tag{1}
p(y_{n+1}|y_{1:n}) = \int p(y_{n+1}|\psi)\,p(\psi|y_{1:n})\,d\psi ,
\end{equation}
where $p(\psi|y_{1:n})$ is the posterior distribution for $\psi$.
Equation (1) applies to both parametric and nonparametric models. The form of $p(y_i|\psi)$ determines the type of model.  
I'll begin with a nonparametric model and then show how it specializes to a parametric model. The nonparametric approach presented here involves an infinite mixture. 
As a preliminary, let $\psi = (w,\theta)$, where $w = (w_1, w_2, \ldots)$ is an infinite collection of nonnegative weights that sum to one and $\theta = (\theta_1, \theta_2, \ldots)$ is a corresponding infinite collection of component parameters. Now let
\begin{equation}
p(y_i|\psi) = \sum_{c=1}^\infty w_c \, f(y_i|\theta_c) ,
\end{equation}
where $f(y_i|\theta_c)$ is a parametric distribution (sometimes called the kernel in this context) where
\begin{equation}
\int f(y_i|\theta_c)\,dy_i = 1 . 
\end{equation}
The prior for $\psi$ is given by
\begin{align}
w|\alpha &\sim \textsf{Stick}(\alpha) \\
\theta_c &\stackrel{\text{iid}}{\sim} H ,
\end{align}
where $\alpha$ is called the concentration parameter and $H$ is called the base distribution. 
The stick-breaking distribution for the weights is characterized by
\begin{equation}
w_c = v_c \prod_{s=1}^{c-1} (1-v_s) , \qquad\text{where $v_c \stackrel{\text{iid}}{\sim} \textsf{Beta}(1,\alpha)$} .
\end{equation} 
In order to implement the model, one must specify the kernel and the base distribution. It is convenient --- but not necessary --- to make $H$ be the conjugate prior for $f$. For example, we could have
\begin{equation}
f(y_i|\theta_c) = \textsf{N}(y_i|\mu_c,\sigma_c^2) ,
\end{equation}
where $\theta_c = (\mu_c,\sigma_c^2)$. In this case, $H$ could be the conjugate normal-inverse-gamma distribution for $(\mu_c,\sigma_c^2)$. 
A parametric can be seen as a special limiting case. Note that $E[w_1|\alpha] = (1+\alpha)^{-1}$. Therefore, in the limit as $\alpha \to 0$ the distribution for $w_1$ collapses to a point mass at one and the model reduces to a parametric model:
\begin{equation}
\lim_{\alpha\to 1}\ p(y_i|\psi) = f(y_i|\theta_1) ,
\end{equation}
where $\theta_1 \sim H$. 
As it stands, the nonparametric model is a Dirichlet Process Mixture (DPM) model. It will prove useful to generalize it slightly by putting a prior on the concentration parameter so that 
\begin{equation}
p(\psi,\alpha) = p(\theta)\,p(w|\alpha)\,p(\alpha) . 
\end{equation}
This will allow the data to help determine the value for $\alpha$. In addition, we will obtain the posterior distribution:
\begin{equation}
p(\psi,\alpha|y_{1:n})
\end{equation}
The predictive distribution $p(y_{n+1}|y_{1:n})$ depends only on the marginal posterior $p(\psi|y_{1:n})$. 
We can use the marginal posterior for the concentration parameter $p(\alpha|y_{1:n})$ to help make a comparison between the parametric model (the restricted model) and the nonparametric model (the unrestricted model). Let $M_0$ denote the model subject to the restriction $\alpha = 0$ and let $M_1$ denote the unrestricted model. ($M_0$ and $M_1$ are just model labels and do not signify "null" and "alternative" models or hypotheses.)
Bayes rule says
\begin{equation}
p(\alpha|y_{1:n}) = \frac{p(y_{1:n}|\alpha)\,p(\alpha)}{p(y_{1:n})} ,
\end{equation}
where
\begin{equation}
p(y_{1:n}|\alpha) = \int p(y_{1:n}|\psi)\,p(\psi,\alpha)\,d\psi 
\end{equation}
and
\begin{equation}
p(y_{1:n}) = \int p(y_{1:n}|\alpha)\,p(\alpha)\,d\alpha . 
\end{equation}
Rearranging Bayes rule produces
\begin{equation}\tag{2}
\frac{p(\alpha|y_{1:n})}{p(\alpha)} = \frac{p(y_{1:n}|\alpha)}{p(y_{1:n})} . 
\end{equation}
Equation (2) holds for all values of $\alpha$, including $\alpha = 0$ (at least in a limiting sense). Therefore we have
\begin{equation}\tag{3}
\frac{p(\alpha=0|y_{1:n})}{p(\alpha=0)} = \frac{p(y_{1:n}|\alpha=0)}{p(y_{1:n})} . 
\end{equation}
The left-hand side of (3) is called the Savage--Dickey density ratio. It is the ratio of the posterior density for $\alpha$ evaluated at $\alpha = 0$ to the prior density evaluated at the same place. The right-hand side of (3) is the Bayes factor in favor of the restricted model relative to the unrestricted model. In other words, by examining what happens to the density of $\alpha$ at zero, we can see whether and how much the data favor the restriction. 
You should be aware that many Bayesian practitioners frown on model comparison of this sort, which amounts to a sharpe hypothesis test (because $\alpha = 0$ occupies a set of measure zero in $[0,\infty)$). They correctly note that the Bayes factor in favor of the restriction can be extremely sensitive to the prior for the concentration parameter, $p(\alpha)$. This sensitivity may increase with the sample size $n$. The reason is that the posterior density near $\alpha = 0$ may become less sensitive to the prior as the sample size increases. The upshot is that one should be thoughtful about the comparison. [BTW, unless I have a reason to deviate, my preferred prior for $\alpha$ is $p(\alpha) = (1+\alpha)^{-2}$. The median of this distribution is one and the mean is infinite.]
As a final comment, let me note that I have not touched on how to estimate the DPM. There are a number of ways to do this. My preferred approach is to convert the infinite sum into a finite sum where $m$ is the upper bound ($m$ should be large enough so that there are alway a number of "empty" components). The prior for $w$ then using the truncated stick-breaking prior, which sets $v_m = 1$. In addition, as is typical with mixture models, it is convenient to introduce latent classification variables, $z_{1:n}$, where $z_i = c$ indicates observation $i$ is classified with cluster $c$. In this form the model is quite standard. See for example Bayesian Data Analysis (3rd edition) by Gelman et al. 
