CE-680 Captain reported a NMAC while on an ILS approach with another aircraft landing on the parallel runway. After complying with a TCAS RA; the flight crew executed a missed approach and returned to the airport for am uneventful landing.

Narrative

We were cleared for the ILS XXR into ZZZ and told to contact tower. Upon initial contact; we were given traffic advisory of a Cessna in the left hand pattern for Runway XXL. We were still in IMC conditions and made that known to the controller. He said he would keep us advised. When we broke out of the clouds the Tower Controller called out traffic again that was turning final to the parallel runway. My co-pilot finally saw the plane and I picked it up a few seconds later. As we were descending on the glide slope I observed the Cessna getting closer to our extended centerline. I know the runways are fairly close together but this was starting to get too close. I asked tower if this traffic was for Runway XXL and that's when the Cessna disappeared underneath us. I called for a go around since I lost sight of the aircraft and we immediately got an RA to monitor vertical speed. Tower instructed us to fly the published missed approach procedure. The flying pilot executed the missed and started a climb which is a slight left turn. This is where we received another RA for a plane in the left crosswind turning downwind. This traffic; which was in sight; was never a factor and we proceeded to fly another approach without incident. I estimate that the first plane that caused our go-around was about 200 feet below us. This was indicated by the TCAS displaying -2 according to my co-pilot. The winds on the surface were out of the East about 11 knots. I'm guessing that the pilot of the Cessna didn't plan accordingly and was pushed through final for their runway. A recommendation I would make would be to add a feasibility note saying with an easterly wind and a busy traffic pattern for the parallel runway to use extreme caution when on final to Runway XXR at ZZZ. This was as close to a mid-air that I've experienced in 35 years of flying.