Is it possible to set up more than one TTL connection? The input that the TTL will be sent to can take two different pulses-one to start/stop data collection and one to trigger an event marker in the data. This requires 4 total connections-the two TTL connections and an analog ground for each. The thread you posted mentioned 2 connections (1 TTL/1 ground), I'm wondering if I can do more.
No problem at all. I'd use the data bits 0 and 1 in your parallel port. This would be pins 2 and 3 on your parallel port. You could use pins 18 and 19 as ground connections as well.

Addressing these separate pins is a matter of sending out the proper byte to the port:
Bit # 7-6-5-4-3-2-1-0
----- X-X-X-X-X-X-X-X <= port is 8 bits wide

Sending decimal value '1' will activate bit 0. Sending decimal value '2' will activate bit 1. Sending '3' will activate both.

Finally, what type of wires/connectors are needed to make this connection?
You'll probably just need a DB-25 connector of the proper gender (remember, males have pins and females have sockets) to match your computer. Then I'd just use some basic 20-22 gauge hookup wire to make the connections.