Flight Instructor with student reported during takeoff roll the aircraft yawed to the left due to jet blast from an aircraft crossing a parallel runway. Flight crew corrected back to centerline and continued takeoff.

Narrative

I was an instructor with a pilot working on instrument currency. We had been cleared to taxi north on [Taxiway] 1 to [Runway] XXR from the GA West Ramp. Behind us was an A319 that had just landed on XXR and was taxiing north on [Taxiway] 1 going to FBO (on the northeast corner of ZZZ). We continued all the way north on [Taxiway] 1 to hold short of XXR; and the A319 turned east on [Taxiway] 2; crossed XXR and was holding short of XXL on 2... pointing east between the two runways at idle power.Shortly thereafter; we were cleared for takeoff on XXR. At about 50kts; and a few hundred feet from crossing behind the A319; the A319 received clearance to cross XXL. He brought his power up to start moving at the same moment that we passed behind him. It yawed our airplane to the left and both the Pilot Flying and myself pushed on the right rudder to get back to the centerline. The event was over and stable in just a couple seconds. We continued for a normal takeoff.I did not think it was of great concern to pass behind the A319 as long as he was at idle and we had good forward speed. However; when he powered up to cross the left runway; it created a potentially bad situation.I think we could have queried the Tower about the status of the A319 when accepting our takeoff clearance; but I also feel the Tower should not have cleared the A319 to cross (thus requiring him to push his power up) with us about to cross behind him.After takeoff; I mentioned things to the Tower who apologized for the situation. After landing I also spoke with the FAASTeem Manager at the FSDO as well as the Tower Manager. I was told this operation is technically legal (which is a bit of a concern) but that it would be discussed with the controller.