1992 Camry occassionally doesn't start

I have a 1992 Toyota Camry. About once every 15 to 20 times I turn the ignition, the car won’t start. Sometimes there’s just a click and then nothing, and sometimes all the lights on the dashboard will flash for a second and then nothing, in each case the starter does not turn over. However, I discovered that instead of needing a jump from another car, if I just connect the jumper cables to the battery and leave the other end of the cable on the ground, the car will start right up after this happens. One other note, the clock is always reset after this. What is the problem and how can this be fixed? Thanks.

Most likely your battery cables are corroded. Take them both off and clean them with a wire brush. The one made for battery terminals works quite well.
All parts places sell them:

Your starter is going out. Most likely it’s the solenoid. Replace it. Testing it probably won’t help 'cause it’s still intermittent. My son’s Corolla did the same thing.

I think it is far more likely to be the connections to the battery or possibly the battery.
Remember, after he clamps some booster cables onto the battery, it seems to work - that points to a bad connection. Also the clock is always reset after this happens. A bad starter wouldn’t reset the clock but a bad battery connection (or bad battery) would.

Unfortunately, older Toyotas are notorious for needing starters every few years.

The starter is actually made by Denso, and the contacts are the real problem.

Your clock is resetting. It’s losing B+ voltage. That has nothing to do with the starter. And while starters are common, specifically the contacts get work out, and still MAY BE, the most likely culprit is the battery itself losing a connection between the cells. They’re just plates that are connected in series and sometimes they lose or have intermittent connection problems. Next time it happens, before you jump it, take a volt meter and read the battery voltage. There are 6 plates at 2.1 volts each (ideally). So, it you read 6v (or anything less than 12 volts), then you have an open connection/cell inside the battery. Corrosion should be cleaned if it’s there and connections should be tight. Check the easy stuff first. so, 1. check corrosion and cable connection 2. check voltage at battery when it happens. 3. If all looks good, look into the starter (but I highly doubt it with the described condition) OR insert a known good battery and try and duplicate. Or buy a new one.

Also, the negative cable from the battery grounds to the transmission or body (not sure on that model), check for corrosion and/or loose connection where the body ground point is.