TMS is an input only signal to the FPGA. The 250 ohm reading suggests that chip is blown. I would remove the 10K pulldown and recheck the resistance. If the value goes up then perhaps the pulldown is the wrong value. Otherwise, I suspect the input protection diode in the FPGA is defective. Nothing you can do other than replace the chip, or configure this device with EEPROM only.
Damaging devices like this is not unheard of using a parallel port cable. I think the problem is making connection to the JTAG pins, but having the ground open or floating. Does your download cable have series resistors in the JTAG lines? This is often done to limit the current on input lines due to misconnections.