If you borrow a car from a different state where does the car need to be registered? Where it's parked or where the owner lives?

If you are just borrowing a car, you don't need to register it in a new state.

You generally register the car where the owner lives. However, most, if not all, states actually allow non-residents to register cars. Generally, the car needs to be primarily parked in the non-resident state.

Owner, owner, owner. Every time, no exceptions.

You said state, so I'm guessing (reasonably) that you live in the United States. In every single state, the owner is legally responsible for their vehicle and any loss or damage it causes. That means only the owner can legally register and insure it.

Legally, a vehicle can only be registered / plated / insured in the jurisdiction (state, province, whatever) in which the owner lives, no exceptions. The name and address printed on the ownership (title), registration documents and insurance documents must all be the same.

All states and provinces have agreed to accept each others' registration and insurance as valid, so there's no problem with out-of-state/province driving. As long as the driver has a valid licence (because duh) and the owner's permission, that driver is legal to be on the road.

If an accident happens, the damage is entirely the owner's problem because the owner is legally responsible for that stuff. If I lend you my truck and you crash it into someone's house, you didn't just have an accident. I did, because I'm the owner. You might have a charge or two laid against you, and that's your problem, but all the damage bills and lawsuits are my problem. And if it goes through insurance, my policy (not yours) will cover it and not yours.

NOTE: Whenever you drive someone else's vehicle, make sure it's got valid registration, plates and insurance, and make sure there is printed proof of that in the vehicle before you drive it. Because if you don't have all of those, you can be charged for driving without them. And those charges are entirely on you, so make sure.

Owner lives

Where ever the owner lives is where he would register his car.

The car needs to be licensed and insured in the state where the owner lives.