If your parked car was hit by an at fault driver and filed a claim with your full coverage insurance will it go up?

Yes, anytime you make a claim to YOUR insurance, it will go up. That's how they get their money back. If you don't submit a claim to them, then no.

I do not necessarily agree with the answer above. What you did not indicate in your question was did the person At Fault have insurance. If they did then even though you filed a claim and your carrier paid you, they would likely recover from the at fault person's insurance. In this circumstance you insurance should not go up.