What happens after I hit a car that was parked illegally?

If you have the required liability insurance, turn it over to them. If not, you will have to deal with the owner. Did the illegal parking have something to do with the occurrence? If so, that can be used to compromise the claim that the owner would make.

You and/or the other motorist could have civil and/or criminal liability. A report to D.M.V. is presumably required. What happens next depends on whether the collision is made the subject of a civil claim or criminal report.

It does not matter that it was illegally parked, unless it was parked in such a way that you could not have seen it. You are negligent in running into it. It is the same as if the car was speeding from the opposite direction and you tried turning in front of it. You did not drive safely so you are at fault.