A few supportive points...
Each parallel card has eight bits you can use. They are not perfectly easy to handle, which you have noticed if you use all of those five one usually get to at first.
And there is accommodation for three such cards in the standard PC.
On top of that there is the usual eight bits going out (to the printer head).
So with a most ordinary computer you could handle 24 trigger points.
But remember, if you want anything near correct timing you have to get the old MS-DOS operating system and that with no "multitasking" types, like Windows, anywhere near that hard disk.
I would be careful depending on magnetic reed switches if using the motors for triggering. Direction, placement and strength make for a very unreliable way to trigger.
The possibly best way, if you only want to catch that a car, any car, is passing is to use an IR-barrier stretching across the track itself.
Remember that you could probably supply the power needed directly from the, digital, track.
Installing an IR-barrier inside the slot itself is not as easy as it sounds. You quite literally need to cut a whole right through the surface to get it far enough up to actually catch a passing guide flag.
Anyhow, good luck!
It is always nice to see projects adding to the hobby!