Car starts in ACC position?

Was it doing this before the new parts? Does it have anything like an aftermarket alarm or remote start? From the description it sounds like either the switch connector or wiring is shorted or perhaps the start relay is fused shut.
To start testing we will pull the start fuse from the fuse block under the left side of the dash and see if it still does it. If it stops then I would look at the connector to the switch real close to see if one of the pins got bent and is shorting in the connector.

No rush, update us as and when you can.

Okay, so you pulled the fuse, and it stopped trying to start in ACC? In that case take a test light connected to ground and go to the start fuse again. Turn the key to ACC and then touch both of the contacts where the fuse was. Watch the light, now turn the key to start and hold it and probe again. If the light comes on in ACC there is a problem in the switch or its wiring. It should only come on in start. You said the switch was replaced? Was that for this same issue and is the switch OEM of aftermarket?

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

Okay, you replaced the key cylinder and the ignition switch and after sometime this issue developed. Was anything else done? Like an aftermarket alarm or stereo? The reason I ask is that based on pulling the start fuse stopping this, there has to be some spot where the power from the accessory circuit is bridging to the starter circuit. It could even be two wires rubbed together in the harness but only through testing will we track it down.

Okay, based on everything so far, the issue isn't in the starter or the starting circuit itself. It has to be in the switch or the wiring harness. The fact that it stopped doing it but then it came back also suggests it's in the wiring. What it sounds like is that power is jumping from the acc/run circuit side of the switch to the start circuit. The problem could be right in the harness near the switch. Another question, when you turn the key and the starter turns the engine over, does the starter stop turning if you move the key to the run position? If you cannot tell, you could remove the SNSR fuse 3 in the fuse box under the hood, that would disable the fuel injectors so that the engine won't start. If you turn the key to ACC and it keeps trying to start in the ACC and ON then it's a short in the wiring.

Nope. If they were the cause it wouldn't respond to the key or fuse and it wouldn't be intermittent.

Okay, so you turn the key to on and the starter spins and stays spinning unless you shift it. Still not the starter or starter relay circuit but it might be easier to find the fault knowing the starter stays engaged.