I like what Soldier said. I got shoulder pain too when doing dips and I'm plenty strong enough to do them with BW. Leaning forward and tucking feet did help, but it still hurt, so like Josh and others I just stopped doing them. There are plenty other ways to do the same exercise.
However, I have come to realize the dip bars I was using were spaced too far apart for me. They where parallel and not angled, so you just had to do them spaced the way they were. I think they were too wide for my build which led to a bad angle on my shoulders and elbows.
Playing around with my sawhorses at home I found I could do them without pain when they were in closer.
Just something else to think about if you're trying to do dips on parallel bars.
I am Anton Zdravko Martin!
Best meet lifts: Sq 150 Kg (330 lb), Bench 120 Kg (264), DL 160 (352) @89 Kg (197)
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