This is a V6 with 173,000 miles on it which I inherited from the ex-wife’s brother a few months ago. Replaced the starter and it was running fine until recently. Now, every once in a while for no obvious reason I try to start it and nothing happens. This is after I have started it and run into town, run some errands, started it a few times with no problem. Come back, turn the key and nothing. There is juice to run all the other electrical systems - lights, radio, dash, fan. But when I turn the key to start it, silence. Then I get a jump from somebody (getting pretty good at wandering around parking lots, cables in hand, begging for help) and it starts immediately and will start the next few times I try. Then the whole cycle starts again.
Your vehicle has the side-mount battery terminals. And on the positive terminal corrosion can form under the red rubber cover causing a voltage drop. Remove both battery terminals from the battery, and on the positive terminal peel back the red rubber cover to expose the terminal connections. If there’s a small amount of corrosion under the cover the terminals can be cleaned up to establish a good connection. But if there’s a lot corrosion under the cover it’s best just to replace the positive battery cable assembly.
If it were the park/neutral safety switch, a jump start wouldn’t make a difference. The switch still doesn’t supply power to the starting system. But if you connect jumper cables to a battery this can be enough to move the positive cable assembly where a good connection is established.
It seems like a battery connection problem or possibly a battery on the fritz. How old is the battery? If it is more than 5 years old, that should be a suspect.
Also, this could be the starter or the starter selenoid problem. When you get a jump start, the voltage from the running car provides is about 14.5 volts, which is more than even a new battery supplies. This extra power could overcome the effects of a bad starter. For a while at least.
Make sure the battery and batter connections are good. Then make sure the connection to the starter and selonoid is good. If that doesn’t fix the problem, consider the starter/selenoid as a remaining suspect.