One of the things to be careful of when using TTL-level signals is that 'low' is not 'off'. Low is typically 'anything less than 1.3 volts', which will often leave you with a measurable current draw even if your relay does not trip. The serial port goes from +12 to -12, and so an LED behaves as a diode, passing zero current when your bit is 'off'.

The parallel port is indeed a great place to tie digital signals; but it is HIGHLY recommended that you use an optoisolator there as well. Most ports are implemented on the motherboard these days, and motherboards still cost a bit more than a bucket of sand. :D