So the parametric term is again the beta ' x and the non-parametric is the kernel, right?
Correct. It's called "nonparametric" because, rather than trying to fit the trend for $T$ using fixed terms like a polynomial or a spline (or cutting it into bands), you're letting the data determine the fit for you. In practice, you do still have some parameters which control how the trend is to be fitted: the bandwidth of the kernel, in this case. A large bandwidth means the fitted curve will be fairly smooth, so it won't be heavily influenced by random noise. The tradeoff is that it may oversmooth the underlying trend, meaning it could miss points where it changes slope or curvature. Conversely, a small bandwidth is more likely to pick up such features, but will also be more strongly affected by random noise.
In principle, any continuous variable that you fit in the parametric part can be replaced by a nonparametric fit. For example, you could fit a trend for age, or income, the same way as you do for $T$. This lets you capture nonlinear behaviour in a data-driven way, ie you don't have to specify what sort of behaviour in advance. You could also handle this problem by cutting the variable into bands and modelling it as a nominal variable, but this is a statistically less efficient way of doing things.
These models are also called generalized additive models, a term which I think is more common than semiparametric modelling.