Generally a car must be registered in the state where it is normally or usually garaged or parked, its "home base". If the car stays in Texas it should be registered in Texas, but if the car is with you in PA it should be registered in PA.
One reason for this is that insurance coverage is based on the address where the car is normally kept. If the rates are lower at the place in Texas where your father lives, having it registered there could constitute insurance fraud. I once knew a person who registered his car at a family member's home in a different part of the same state where traffic was lower and insurance rates were less. When this came out he had to pay the difference in insurance rates plus a fine.
Most states require that you register a car within a limited time after moving into the state with it. According to this official page:
All new residents are required to make application for Pennsylvania title and registration of their vehicle(s) within 20 days of establishing residency in Pennsylvania.
Pennsylvania title procedures require that the out-of-state title be surrendered to the Bureau of Motor Vehicles when applying for a Pennsylvania title. The current lienholder of an out-of-state title may require that the vehicle owner refinance the balance of the outstanding lien through another lending institution and pay the out-of-state lienholder in full. A Pennsylvania Certificate of Title will only be issued when the out-of-state title is submitted with the completed Form MV-1, "Application for Certificate of Title."
If you are titling and registering a leased vehicle, the leasing company will also be required to complete Form MV-1L.
Thus you must notify the credit union and the relevant insurance company of the new address of the car. Since one co-lessee is a member of the CU, they probably will not require a refinance, but that is up to the CU and its policies.
There should not be a great deal of paperwork involved. I have moved with a car from one state to another, and it was just a matter of submitting the registration form for the new state.