B737-700
Flight Crew Air Transport Pilot (ATP)
Conflict Airborne Conflict
Flight Crew Took Evasive Action
Procedure
We were on the Tip-Toe charted visual approach to 28L SFO; 5 miles behind a Heavy B767. ATC had slowed us to 160 KTS approximately 10 miles from the runway to keep our spacing from the B767. Inbound from the San Mateo Bridge; we slowed to our final approach speed (approximately 130 KIAS); and we were on the localizer and glide slope. We were fully stabilized approximately 1;300 feet AGL; due to our slow speed crossing the outer marker. A B737 was approaching behind us on the Quiet Bridge visual approach to the parallel 28R. It is intended that aircraft stay staggered on these close parallel approaches. When we were around 1;000 feet AGL we got a TCAS Resolution Advisory which said 'descend'; and displayed command bars. I followed the command bars while my First Officer looked for the aircraft. He immediately saw it; and said it appeared he was going around. I was slightly below the glide slope and began to correct; now that we had the intruding aircraft visually. I also moved slightly left of my localizer to increase lateral separation. Then that aircraft lowered his nose rapidly to recapture his glide slope at which point he overtook us rapidly. I just made sure we maintained lateral separation while he passed us. He touched down well in front of us on the parallel Runway 28R. We landed normally on 28L. I have never had another aircraft overtake and pass me on this or any other close parallel approach. It was shocking to say the least. Any passengers looking out the window must have been stunned to see how close the other aircraft was to us. I can't control what other aircraft are going to do in any given situation. In this case I cannot go-around with a descend command from TCAS. Even when we had him visually going around would have been more dangerous in terms of collision. Any lateral movement during the go-around procedure could have been disastrous.  I felt it best to just maintain position and let him pass us. Once he was clear; we were still well within the parameters of a stabilized approach; so we continued for a normal landing.
Air carrier landing SFO during side by visual approaches to Runway 28L/R described a passing event; expressing concern regarding the procedure.