A bit more info in regards to overall performance and condition would be helpful, but given the info at hand... How's the temp gauge reading?
Backfires happen for a number of reasons. Generally they happen while running though...
The most common cause in my experience is exhaust leaks. Oxygenated air gets in with the hot exhaust gases that still have a little unburnt fuel, and it'll spontaneously ignite causing a backfire. They're pretty easy to track down using only your ears. This requires the engine to run and is not likely your issue.
Another related cause is a poorly tuned carburetor. Running too lean (a lot too lean) can cause the mixture to not be burnable in the cylinder. It goes out the exhaust and ignites when it comes in contact with the hot gases from cylinders that did burn. This will cause a lot of additional issues that will out it pretty easily. Again, must be running, you'd be crawling for lack of power, and it would run pretty badly.
Running too rich can cause backfires in a similar manner, though this usually needs to happen in connection with the aforementioned exhaust leak.
Crazy timing issues can cause intake backfires. These are usually pretty loud and dramatic... Crossed spark plug wires are typically more culpable than anything... It's hard to actually be off that far with your timing and still run. That said, really bad spark plug wire insulation can do weird things if things get crossed. Things arc in weird ways sometimes. Again though, gotta be running.
Finally, your most likely scenario: Hot spot/dieseling. Something is causing your engine to run way too hot and there is enough residual heat after shutdown to ignite fuel (typically in whatever cylinder was on the intake stroke) shortly after shutdown. This will be an intake backfire, so it'll be a bit more dramatic (a loud sharp pop under the hood versus a more subtle pop from the exhaust). This could also manifest itself in the engine continuing to run for a revolution or two after the key has been turned off. Culprits here include timing being way off or running pretty lean. Timing being too far off will cause the whole engine to be pretty hot as the burn happens at such a time that it is in contact with the cylinder bore versus the exhaust system. Running too lean will cause it to get hot in certain places (i.e. spark plug electrodes, high spots between the valves/spark plugs in the head) and these hot spots can effectively make up for the lack of an actual spark.
Suggested action: check your timing first because it's the easiest. Then pull a spark plug and see what it tells you. It's possible that it requires one circuit or another on the carb, so it may be that around town driving doesn't cause you ailment, or that highway driving doesn't cause it.