A320 Pilot reported cabin pressure control failure on climb and performed an air turn back and landing at departure airport.

Narrative

On initial climb out passing 12;000 ft. cabin pressurization began to run away. The pressurization page came up and we observed 4;000 FPM. We immediately stopped our climb at 13;000 ft. When the plane leveled off and the power came back the cabin altitude and vertical speed stabilized. We had no ECAMs or warning or cautions. We asked to descend to 12;000 ft. Cabin altitude was operating normally. We asked to climb to 15;000 ft and when the power came up and we began to climb the same happened.We immediately ask for a descent to 10;000 ft. Again the cabin altitude and the rate of climb/descent was normal. We elected to [request priority handling] asked to hold at 10;000 ft. to burn fuel so as not to have an overweight landing. The cabin altitude stayed at 900 ft. during the hold. On descent for the approach the jump seaters noticed the cabin alt go into a slight climb for a few minutes than return to normal. [We had an] uneventful landing. [The problem was] high pressure air and pressurization computers malfunctioning [due to it being an] older aircraft. Some of the aircraft are old. Mechanical malfunctions will occur although in my years with the Airline they are very rare.