A military jet trainer pilot reported a NMAC with a corporate jet turning final to the parallel runway.

Narrative

This is based on my recollection only. Factual information should be gathered through tapes. On a stabilized Visual Approach on extended centerline to ZZZ; approach switched us to tower at just outside 10 miles after we called the field in sight. Upon check-in with tower we were cleared to land on [Runway] XXR and given caution due to wake turbulence of a heavy B767 on 2-mile final. We saw the B767 easily on approach to XXR and we were 8 plus miles in trail. A few miles later as we began our descent; tower asked us if we could take a turn to initial. This confused me because we were on a straight-in stabilized approach and cleared to land behind the B767 which I could see taxiing off the runway. I told tower we could if needed however were not requesting the overhead because we were on a visual approach for a full-stop.At that time tower directed me to side-step to [Runway] XXL. As I began to sidestep to XXL; I saw the B767 turning left off XXR at midfield towards XXL which was unexpected since we were now cleared to land on XXL and the B767 would likely not be clear by the time we landed. Simultaneously as I began the sidestep; I got a TCAS alert within 1 mile co altitude and tower made a radio call to a Gulfstream not to overshoot. I immediately looked to the right and saw a Gulfstream belly up to me at a 30 plus degree angle of bank co-altitude converging on my flightpath. I took immediate action; initiating a hard left turn and rapid descent to avoid the Gulfstream. After confirming adequate separation; we executed a climbout and turned tower downwind for an uneventful full stop.