Air Carrier departure from Runway 1R at SFO described conflict event with traffic departing Runway 1L; voicing concern regarding the departure procedure from the closely spaced parallel runways during IMC conditions.

Narrative

Taking off from Runway 01R SFO. Fog and low clouds had just broken up overhead the airport; but was still over water just off the departure end of 01R. We were cleared for T/O on 01R and immediately after that a corporate jet was cleared for T/O on 1L. At about 200-300 ft AGL; we entered IMC conditions due to low clouds off the end of R/W. The SFO 8 STAR requires a 030 heading after T/O; so at 400 AGL the F/O started a right turn to 030. We broke out on top of low clouds at about the same time. The F/O exclaimed 'whoa' and I leaned forward in my seat to see the corporate jet at our 10:30 position and very close; 300-500 ft. horizontally and about 100 ft. above us. Too close for comfort. We continued our right turn to 030 and separation was restored. I think that simultaneous departures off 1L and 1R are questionable in VMC conditions; but are unacceptable when IMC conditions exist or are expected. I think staggered departures should be required when IMC conditions exist in the immediate area. Also; the tower should advise departing aircraft of each other's proximity on close runways like SFO 1L and 1R.