There's a lot of competition from wreckers who've discovered Ebay, and I think the trend would be more competition to come.
Shipping large items, like body panels, hoods, trunks, etc, will cut deep into your profits unless you do enough volume to get a volume discount with a shipper. It's not easy packing that stuff, either.
Craigslisters will eat away at you in one way or another. They won't show when they say their going to show, and some will take bargaining games too far. They'll text you at all hours, the string you along for a long time only to buy one tiny inexpensive item, or nothing at all.
You will need a lot of room, a place to store stuff out of the rain, including the car itself when windows and trunk lids start to sell.
You'll need to properly deal with fluids, gasoline, brake fluid, power steering fluid, radiator fluid, engine oil.
You better have title stuff in order, or you're gonna get in trouble sooner or later.
The things that sell the fastest are things that are most commonly damaged in fender benders, accidents, and vandalism - tail lights, turn signals, head lights, and trim.
Certain cars will have demand for certain items. For example for the Toyota Celica GT I parted out, trunk parts were in demand, because that particular model in the early 90s had issues with the trunk hinges where they mount to the trunk.
Every car has things that break easy on it, so those parts are in demand, however the vehicle you get is also going to be subject to those issues, so unless you're out of the norm, you're going to have that part to offer up.
Try to find out from sellers what they had replaced or serviced before they sold to you. You'll have a selling point with new or relatively new parts. Very often, when you buy an older car, several parts are new or relatively new, like radiator, brakes, tires, starter, alternator, etc.