It could be ground or power. The only way to determine is to get on the socket terminals when they are not working and see what is missing +/-. Is it ALL markers or just one? Tailighs also come on with the parking lights on. Are they working?
The side markers have no dedicated ground. They ground through the opposite + lead and filament in the front fender lamp (it's known as a "floating" ground). If your front fender lights are staying on continuously, it's most likely a bad connection or corroded lamp socket in your side marker light, or could be the same in the front parking/turn signal connection/socket.
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