Starter not working

what transmission, here. automatic or manual stick shift?
S10 not blazer S10

old cars this old (pre digital documents) are hard to find and when found are not complete. or poor .
A/T box?
the PRNDL switch must be good. and cranks in both Park and neutral.
yes the relay clicks too, if not the PRNDL is prime bad part.
there is fuse named CRANK.(1st inline here)
the starter can be hot wired (a test) to prove starter work.
if stick shift TRANNY it is called the clutch switch. to crank,
some books say hood up fire wall, center is relay. IDK


see why we ASK transmission first/
if the relay does not click the fuse crank blown
or PRNDL is dead only on P mode, try N.
if stick shift,, clutch CCP switch bad or just mal-adjusted.
ST below goes to the relay then to starter solenoid
the relay colors are there. Green or Purple.

PRNDL is on side of TANNY part of shift linkage.

improper install -- defective -- wasnt the problem

I looked at 04--5.4L engine.

Check the fusible links at the battery and at the starter.Just pull on the red wires and if one comes apart easily it's a blown fusible link.

Try makking sure both battery terminals posts are clean and connected good and the battery is fully charged failing that tap the starter motor with a hammer while someone else tries to start it, here is a link to module locations on a 92 likely the same
https://www.justanswer.com/buick/813lr-i-m-working-1992-buick-park-ave-need-know.html

Did you check for power at the starter ignition terminal? It is the small YELLOW/BLACK wire on the starter. If there is power there, replace the starter. If there is no power there, you should check your starter relay an its associated fuses.
The relay primary side is protected by fuse number F2.31 (15 Amp) in the Central Junction Box. This fuse provides power to the clutch safety switch on the WHITE/PINK wire. Then the power leaves the clutch safety switch on the TAN/RED wire and goes to pin 86 on the relay.

The relay secondary side is protected by fuse number F2.113 (30 Amp) in the Central Junction Box. It provides power at pin 30 on the relay, which is also located in the Central Junction Box.

Pin 87 at the relay is the YELLOW/BLACK wire metioned earlier and goes to the starter motor.
Pin 85 at the relay is the primary GROUND. This should be ground at all times.

The ENTIRE starting system can be checked/diagnosed from the pins at the starter relay if you use the correct methods and equipment.

the neutral switch is located in the left side of the transmission and its connected to the shift cable.

Start by checking all of the fuses and all of the fusible link wires, and you will know a fusible link wire when you find one because they will stretch similar to a rubber band. Fusible link wires are usually located at the fuse/relay junction box in the engine compartment, or at the starter solenoid, or sometimes at the fire-wall. Also check the fuse/relay junction box for any PAL type of fusible links.



A Typical Fusible Link Wire.

You have no power from the switch to the starter. Check all you wiring that goes through the fire wall and look for burnt connections or just bad connections.

Check for a burnt fuse link Not the fuese but a fuse link, look at the positive cable going to the battery. Sholud be other wires hooked in the rubber end cap. These other wires are fuse links. feel them for a break or use a test light and probe them for power, repair the one thats burnt.