I am not exactly sure what you are asking here, but I think that is more a terminology disconnect than anything. I haven't even heard someone say Vapnik-Chervonenkis dimension in a long time, so we may be coming at this from two different standpoints. But I'll try to answer as best I can.

When you say parametric regression model, do you mean something like s(x) = a + bx where the parameters here are a and b? If that's the case then I might say that a GP model has N parameters, where there are N pieces of data. You could argue (and I think Rasmussen does) that GPs are nonparametric, since there are N parameters and N pieces of data, but I don't usually use that term about GPs.

As you point out, you do not need to carry around the data after you have fit something like a s(x) = a + bx model ... you only need a and b to make predictions. This is not true for GPs, where you will always need the {x_1, ..., x_N} points in order to make predictions. The benefit to using these kernels as your basis s(x) = c_1K(x, x_1) + ... + c_NK(x, x_N) instead of s(x) = a + bx + cy + dx^2 + ... is that there is a guaranteed unique interpolant to your data. It would depend on how you define "error" (and this is a constant issue in any sort of real-world setting) but if you are defining error by how well your model fits the data then a GP model would perfectly fit any data. I'm not saying that is really what you want (I talk in http://blog.sigopt.com/post/132959177928/sigopt-fundamentals... about why that is a bad idea at times), but I am suggesting that there are certain properties that GPs satisfy very nicely.

I would need to understand why you are saying that the number of parameters you are fitting is much smaller. I don't immediately know how you can guarantee that just by looking at a bunch of x and y values. If there is a specific reason you think your data came from a very simple function, then that would make sense, but if you can't make that assumption I don't know why it would immediately be preferable to use one model over another.

Given that, if you do think that your data came from a relatively low parameter source (to put it in tangible terms, suppose you have 20000 pieces of data and you think you should only need 50 components in your s function) then you can create a 50 term model using reproducing kernels instead of polynomials (or whatever else). This is often called an RBF network (or kernel network). You can find a great reference online (http://www.eng.buffalo.edu/~psingla/Teaching/MAE502/ReadingP...) or a decent reference in chapter 18 of my book (kernel-based approximation methods in Matlab). If you choose to use this strategy you would be creating a model where you would only have to carry around 50 parameters, instead of the 20000 that you would have to use in the GP setting. The benefit over polynomials would be flexibility and locality, the benefit over GPs would be cost (50<<20000 so predictions are cheaper). There are also Support Vector Machines/Regression which do this stuff in a sparse framework (less that 20000 basis functions) using reproducing kernels.

Basically, I would totally agree with you that simpler is better. A the end of the day, I think of GPs as a useful tool for identifying local behavior, but I try not to use them globally (although in this paper I looked at the global limit, http://epubs.siam.org/doi/abs/10.1137/110824784). Probably the best strategy, if you believe that your data comes from a low complexity function and should only need a few parameters to fit it well is to explicitly define a nonzero mean for your GP; see (Chapter 2.7 in http://www.gaussianprocess.org/gpml/chapters/RW2.pdf) or Chapter 17.3 in my book. This lets the low parameter model handle most of the heavy lifting and the GP handles the final touches. You still have to carry around the N pieces of data (sorry) but you get closer to that parsimonious result that you wanted.

Incidentally, if you're just worried about carrying around more data than you want to, or that the cost of making predictions is too high, there are computational techniques to deal with this. Two good friends of mine, Lei Wang with tree codes (http://epubs.siam.org/doi/abs/10.1137/120903002) and Jie Chen with H-matrices/semiseparable/recursive rank (http://www.mcs.anl.gov/papers/P5112-0314.pdf), worked on something like this for massive problems (I think Jie Chen was in the billions while we were at Argonne together). So there are options out there even for those extreme cases, depending on what your situation/goal is.

Hope that helped. Thanks for reading.