This is almost certainly due to cooling. Serving on heated plates will buy you some time. If (at home) you use microwave plate warmers you can place them under the plates after you've used them and the residual heat will keep the plates hot. You need to strike a balance as you don't want it to cook too much after serving.
Cream thickens on cooking (unlike butter which solidifies as it cools). So cooking the sauce for a shorter time might help as well. In fact the recipe I've got for this cautions against overcooking the cream (without saying why).
You don't say how much parmesan you add in proportion to the other ingredients, but you may be onto something with reducing it. Unfortunately this will reduce the flavour, and it's not like you can replace it with a much tastier cheese.
The other thing that's commonly done is adding a little pasta water to the sauce. If you're stirring it together, this can simply be a matter of not draining the pasta too much (this also depends on the shape). If you're spooning it over the top, taking a little of the starchy water just before you drain the pasta, and adding it to the sauce, could be worth a try. Again I don't know your exact quantities (like how big your glass is) but anyway it's a matter for experimenting. A tablespoon or two might be a good place to start. You probably need to put it in slowly while stirring.