A319 Captain reported they descended in response to a TCAS/RA for parallel runway traffic then received a Low Altitude Alert from ATC.

Narrative

Vectored onto XXL visual into ZZZ using ILS for backup. Paired with a commercial aircraft going to XYR. As aircraft converged to their final approach courses; we received an RA to descend it appeared that the pairing traffic had overshot their final approach course and were drifting into the course for XXL. We received a brief TCAS traffic call that quickly turned into an RA with a descent of 1600 ft./min. Even before the traffic call; I had the First Officer increase there turn rate onto final to avoid getting an RA. At this time; we are already fully configured; checklist complete; and auto pilot was already off. I elected not to turn off flight directors to maintain a reference to the glide slope. The RA took us all the way down to approximately 500 feet where we received a Low Altitude Warning alert from tower. At this point we leveled off and I elected not to go around because it appeared that the traffic was still slightly in our final approach path. As we got closer to the airport the pairing traffic seems to correct and lineup for XYR. At no time did I feel like we would be close enough to that traffic to cause a collision. We continue the approach to an uneventful landing. Weather was VMC light winds and good night time visibility. The initial clearance after the arrival was the visual XXL. However; once being vectored; we were off the visual path and I put in ILS XXL for back up for our visual reference. For most of the approach we were well left of the XXL ILS course. Which is what led me to believe that our pairing traffic was well left of the final approach path for XYR. We maintained visual reference to that traffic throughout the entire event.