In most states in the United States, for many insurers, yes, your credit report is one factor in determining your rate. California, Massachusetts, and Hawaii are the (current) exceptions to that; in those states it's not legal to do so.
The credit check is a soft pull, though, and does not affect your credit score. So the answer to your primary question is no, it does not affect your credit score, but it is affected by your credit score (or more accurately your credit report; the score used is a custom score, not the standard FICO score.)
Your parents' rates may have decreased due to many factors, including your presence in general (particularly if you're under 25), but one of those factors could be your credit history - though it's not necessarily the case that your credit factored into the rate at all. You could ask the insurer if it is factored in, or if only the primary insured's credit is pulled.