Testing proportional hazards assumption in parametric models I'm aware of testing the proportional hazards assumption in the context of the Cox PH models, but I haven't encountered anything relating to parametric models? Is there a feasible way to test the PH assumption of certain parametric models?
It seems like there should be given that parametric models are only slightly different from the semi-parametric Cox models?
For example, if I wanted to fit a Gompertz mortality curve (as below), how would I test for the PH assumption?
$$\begin{align}
\mu_{x}&=abe^{ax+\beta Z}\\
H_{x}(t)&=\int_{0}^{t}\mu_{x+t}\,dt=b(e^{at}-1)e^{ax+\beta Z}\\
S_{x}(t)&=\text{exp}(-H_{x}(t))
\end{align}$$
I suppose in general what I'm asking is: for parametric survival models, what are some ways of evaluating goodness of fit of the model and also testing for assumptions (if any) of the model?
Do I need to check for PH assumptions in a parametric model or is that just for Cox models?
 A: A complete answer depends on the nature of your parametric survival model.
If your parametric model incorporates covariates in a way that the relative hazards for any 2 sets of covariates are in a fixed proportion over time (as your Gompertz model seems to), then your parametric model is making an implicit proportional hazards assumption that must be validated in one way or another. As this answer by @CliffAB points out for the specific baseline hazard assumed by a parametric model:

a Cox-PH model fits a model with A) proportional hazards and B) any baseline distribution. If the best fit with the requirements of A) proportional hazards and B) any baseline is a bad fit, so will a model with A) proportional hazards and B) a very specific baseline.

This would suggest that you first try a Cox survival regression to test proportionality of hazards. If the assumption is violated with the empirical baseline hazard determined by the Cox regression, then there is little point to proceeding with any parametric model that implicitly assumes proportional hazards. If you can proceed with such a parametric model, the R survival package provides several types of residuals for evaluating parametric models with the residuals() method for survreg objects, in addition to the suggestions made by @Theodor. 
If, alternatively, your model incorporates some covariates in a way that provides for non-proportional hazards as functions of covariate values (e.g., different baseline hazard shapes), then there is no need to test specifically for proportional hazards with respect to those covariates. Stratifying on those covariates would allow tests of proportional hazards for covariates that are assumed to involve proportional hazards. You will of course need to test how well the data fit the assumptions of your model, but insofar as proportional hazards aren't assumed (explicitly or implicitly) then they don't need to be tested.
For further background, Harrell's Regression Modeling Strategies devotes chapter 18 to building and evaluating parametric survival models; more cryptic but useful coverage of this topic can be found in examples worked through in his freely available course notes.
A: An easy way is to compare a model with a fixed covariate effect, $\beta$, with an extended model with a time-dependent effect $\beta(t)$, with a flexible function form - for example using splines. 
If the proportionality holds, then $\beta(t) \equiv \beta$, and the two models would be virtually indistinguishable. If the proportionality does not hold, then the model with the time-dependent effect should provide a significantly better fit. 
edit: For the most part, having a parametric baseline does not change things so much in terms of assumptions. As with any parametric model, to test the model assumptions, you must specify a possible departure from the model assumptions. 
One of the strongest assumptions of a proportional hazard model is the proportional hazards assumption; in particular, this means that the effect of the covariates is constant in time. The idea is that you nest the model in a more general model and you compare fits. 
So, to answer your question: you need to check for PH assumptions in parametric models as well. The graphical ways (log-log plots) should work the same as in the Cox model as well. The residual-based methods should work as well, but I am not completely sure of that (I am quite confident that the martingale methods work, since the whole theory applies in parametric models as well). 
