Uncle Ben,
Please post the pump curves and the system curves if you have them.
Little snap shots of data don't help us.
Start with operation of pump A - pressure at the pump, pressure at pump B and flowrate
Then advise pressure when Pump B comes on line.
what is the pressure switch set at?
Pipe sizes and lengths would help plus what the conditions are at the far end? A fixed pressure or pumping into another tank or?
Pump B looks like it is backing out pump A, but normally a pressure trip on a centrifugal is not a great plan when feeding from a tank as you can only have a certain max head coming in.
but yes, if the issue, which is what it looks like, is that pressure drop in the line from pump B down is too high for both flows, then reducing flow from Pump B is a potential solution.
Remember - More details = better answers
Also: If you get a response it's polite to respond to it.