There are garages near the school of art, but they may have fairly high rates/monthly fees. NHV has a lot of people commute into the city though probably less after COVID.

Most street spots are metered and many have time limits. You may be dodging parking enforcement.

I think the risk of your bike being stolen when street parked is equal downtown and in the neighborhoods you mentioned. The risk at home maybe higher when it's dark.

Generally, I don't think you'll get much enjoyment out of a ride from east rock to the art school with traffic/lights/road quality. It's close enough that biking is easy or you could even walk. Yale also has a shuttle system and CT buses have been free (fingers crossed they extend it). You could never drive to school if you wanted.