You can bypass the park/neutral switch by connecting a wire from the park/neutral terminal on the starter relay and grounding that wire to a good ground. This will bypass the park/neutral switch and the wiring associated with it. This should allow the starter to crank. Be careful, the starter will crank with the trans in any gear.

If it cranks with the key with the park switch terminal on the starter relay grounded, the park switch, or the wiring to or from it is bad.

Are these starter relays you are installing all new, or are they known to be good? Today's switches are crap, new does not mean they are good anymore. I installed 4 new headlight switches in a car before I got one switch the functioned correctly. Gene