Hi, starting with the relay driver. You are using the buffer no. 8, with input on pin 8, fine. The output you should use is 11 and it seems like you do.
The COM pin (10) should be connected to +power and you seem to have done that too.
THE GND pin(9) should be connected to power gnd, ans it seems like you have done this. This is crucial.
The ULN2803 is providing GND(!) (from pin 9) on the output, not 12V and the relay thus needs +12V on the other side of the solenoid.
When the ULN is not driving the output is 'floating' like an open wire with just the diodes in revese protecting the transistor from spikes.
The solenoid of the relay is pins 85 and 86. You can keep the 86 connected to pin 11 of the ULN, and connect 85 to power+ (12V).
The switch in the relay is on pins 30 and 87. You can keep the 30 connected to + on power source but move the track+ to the 87 terminal.
The ground on the parallel port should also be connected to GND on the power source.
This setup will activate the relay when the output of the parallel port is high (5V), if this is desired?
Your circuit will never work for 2 reasons: the relay is never activated, given GND on both sides of the solenoid and one contact pin not connected (87).