Wanted to move to Texas and I'm on parole for 5 years. Will my felonies still show? Also, Is there a 3 strike rule in Texas because I have 3 felonies already and don't want something to happen and I go away for life.

Your criminal record is national, so it shows up anywhere you go, It would even show up when ran over seas ( but takes a few days to come back)

You will have to get permission to move from the courts, first from your current state and then from from Texas.
Your sentence will not change after you move.