Google Maps Street View appears to show paid parking in Piazza Re Manfredi, which is between the castle and the Piazza del Duomo.
There are parking signs with the blue P symbol but I can't make out the small print.
Would this mean that that area is open to the public, that it's not in the ZTL?
I found this map of Trani ZTL but not sure if accurate. Seems to cover only a portion of the Trani centro storico.