I’d have to see your plumbing setup (email me some pictures if possible) to answer this question because there are a few possibilities. It appears that the sump pump discharges into the sewer pipe, maybe you also have a septic tank installed…
If it is a septic, it might be filling up with water from the saturated soil (during and after heavy rain), and discharging even more water into it from your sump pump results in a backup int to the sink (the sink could be the closest to the septic).
If you have the city sewer system, it may be also overflowing during the rain and cannot accept more water being pumped into it. There might also be some blockages inside your drain pipe that result in backup with larger amounts of water running through the pipes.
Some jurisdictions don’t allow sump pump discharges into the sewer system, but I’m not sure about yours – I would start from disconnecting the sump pump from the pipe it is currently discharging into (apparently it is your sewer system).
Put an extension on the sump pump discharge pipe and run it as far as possible from the house so the water doesn’t flow back into the sump pump well under the foundation. Give me some more information describing your plumbing connections / waste system.