While a PCI-to-PCIe bridge is possible and is indeed present on some modern boards that have PCI-less chipsets, I am not aware of any company selling a PCI-to-PCIe adapter card and even if someone did make such a product, the card would no longer line up with the IO slots which would make it impossible to secure the card properly and may also make it nearly impossible to plug anything into rear connections.
So your options are:
1) buy a new board that has PCI slot(s) which are becoming increasingly uncommon
2) upgrade to PCIe equivalents
3) ditch the cards