An air carrier pilot reported on initial climb they received a TCAS warning for an aircraft which departed the parallel runway.

Narrative

I was the pilot flying. Cleared TO RWY XXR and told by Tower that aircraft was taking off on XXL was going straight ahead. When we were lining up; the citation XXL wasn't moving yet so I figured he was waiting for us to takeoff first. During our takeoff roll; I looked over my shoulder and didn't see him so I assumed he was smart enough to delay and keep us in sight. We were on the ZZZZZ SID. Following the SID track; at 1200 feet; I looked out left window and saw the Citation off our left wing about 200 feet off our wing. I started right turn to turn away from him. We got a TCAS monitor vertical speed" and then immediately got a "clear of conflict". I saw the target was at our altitude on the TCAS. I continued my turn towards ZZZZZ1 while asking for flaps up. Cause: I am assuming the Citation had us visually but got too close. If ATC wants to takeoff 2 aircraft at the same time; XXR AND XXL; time they should clear the trailing aircraft to maintain visual separation. Didn't get visual of him until he was right off our left wing because he took off after us and wouldn't be expected to catch up. Of course; we were busy with takeoff phase of flight."