Hello. I have a 2002 Acura TL with 228,000 miles on it. Over the past 6 months or so I have had a problem that the car will randomly not turn over when I try to start it. I use the car every day but this happens once every week or two.
I have replaced: Battery Terminal Connectors, Starter, Starter Relay, and Ignition Switch.
Troubleshooting: First I had the battery tested and it tested good. Next, with the ignition key in the run position, if I jump a cable from the hot side of the battery to the solenoid starter control terminal the car will start every time (thus I keep a little piece of wire in the car for that just in case, lol). I have two sets of keys (and FOBs) but the problem can happen with either set. The only other thing I have noticed is when it is cold outside I don’t have a problem. I live in Tennessee and from late December until late February I had no problems starting the car. I appreciate any help.
Since you can jump battery power to the S terminal on the starter solenoid and the starter turns you know this isn't a battery cable issue (or a starter issue).
The place to troubleshoot this issue is at the starter relay socket.
From all the parts you've already replaced this sounds like an intermittent issue with the Transmission Range Switch (neutral safety switch) ((or it's wiring/connector)).
Here's a typical early 2000's Honda starter wiring diagram.

The "Control Side" of the Starter Relay gets its Ground through the Transmission Range Switch.
Here's a typical Honda starter relay but you can read the circuit map on yours if yours is configured differently.

To test this circuit you can cut 2 inches off of that "jumper wire" you have, and in the next "NO Crank" issue you can use it to jump these starter relay terminals.

So attach your "jumper wire" and then KEY IN START POSITION (refer to the above wiring diagram. Both the "Load Side" and the "Control Side" of the starter relay only get Power when the Ignition Switch is in the START Position).
If your Acura's starter spins you know that the wiring between the relay and the "S" terminal on the starter solenoid are good.
Next you want to make sure there's Ground on one of the 2 remaining Starter Relay terminals when the car is in Park and Neutral.
You just need a cheap test light.
One of those (top) 2 sockets will be grounded when the car is in Park or Neutral.
The other socket will have battery voltage with the Key in the START Position.
If you see Power on one but No Ground on the other during a "NO CRANK" then check out the Transmission Range Switch
Once again @jack62 - excellent.
That generation Acura TL is very similar to my 1999 Honda Accord.
When my Accord wouldn’t start, these were the two major culprits:
1) Relay underneath the steering column. I am 99% sure it was a starter relay.
2) Something on the fuel pump, where the whole fuel pump needed to be replaced. It was something electrical, but cheaper just to replace the whole thing.
There were two other instances for me which I do not think is the cause of yours, but just in case:
Loose battery terminals.
Starter.
Check your grounding connections for looseness and corrosion. You have the new parts but it sounds like the circuit is not getting completed, except in warm weather when the parts expand and make contact.
I had a 2004 tl with the same random issue, but mine turned out to be the starter solenoid, which I just replaced the hole starter, and it took care of the problem, but your case seems a lot different as you already changed a few parts and still have the same issue. It doesn't hurt to check the starter solenoid again. In my case, it would also work with a jump directly to the starter as you are bypassing the starter solenoid by doing that and remember that new parts are remanufactured parts, not brand new. Put a test light on the starter solenoid input and see if the light goes on when you turn the key to the start possition but the starter will not crank if that happens replace starter and solenoid, if you get no power to the solenoid trace the wire back until you find the fault unit that is not transmitting signal. Took me awhile since it was an intermittent problem but I ran a wire from starter solenoid to the inside and when I got the no start I just looked at light and it was on meaning I was getting signal to start but solenoid was not powering the starter. The solenoid is attached to the starter, or at least in my 2004 it was and acura doesn't sell it you have to buy the starter. Good luck.