If you use a Freccia or Intercity you need a reservation for a specific train.
For TI / TTPER regional trains your ticket is valid for around 4 hours from the train it was bought for (see recent thread), I don't know the Trenord rules but one leg would be TTPER. Beware the 'check in' message if it applies.
Easiest is to book each single ticket when you are there, either at a station or the Trenitalia app.
Especially coming back, the Frecce from Parma often have come from Bari or Pescara. Last May I used one which was late at Parma and reached Milano C 50' late on my 50' journey. So I'd buy on the spot on the day after checking real time running but with loads of time for fallback before the flight.
If using regionals, only trenitalia.it will show you the Parma-Milano RVs because they are operated by TTPER. Not DB nor any international planner shows them because TTPER times still don't seem to be fed to the wider world.