Sounds to me as though you've been told some baloney as platys and guppies are hardwater fish and corydoras catfish are tolerant of hard water. Some species of synodontis rely on African cichlids for their breeding behavior, and African cichlids live almost exclusively in very hard water, so I would assume synos are okay with these conditions as well. Most plecos are okay in medium-hard water. I know nothing about paroons.
You can buy kits that test the water hardness, usually as one kit that tests both gH and kH. kH is the carbonate hardness, a measurement of the amount of calcium (as calcium carbonate CaCO3) dissolved in the water. gH is a measure of the general hardness, or the total dissolved salts - all of the minerals dissolved in your water. Usually when asking for the hardness we'd want a measure of both.
What are the fish doing? (In what way are they sick?) Judging by the species you have there, hard water would not cause sickness amongst all those fish. Could be an LFS grabbing at straws. If you've only moved 15 miles you are probably still on the same water supply, is this the case? If so it's definitely got nothing to do with a change in hardness, and if some of the old water went into the tank, the fish have had time to adjust anyway. Has the usual pH reading changed between your old house and the new one?