There's little black specks coming out of the bathtub faucet, enough to fill up the tub.I have to run the water for about 5 minutes until it clears up. The leasing office said its old paint from inside the pipes, but I do not believe them. They said its still safe to shower in, but I'm pretty grossed out. They also haven't attempted to fix the problem. I bathe my son in this water, so no matter if its safe or not its still pretty disgusting. Any ideas???

The only way to determine what is in your water to have a professional water treatment specialist or reputable licensed plumber to do an on site inspection; take and test water samples, etc. We cannot diagnose it over the internet.
Are you on city water or a well? If you are on a well, if you take a refrigerator magnet and sweep the black specks in the bottom of the tub, do they stick to it?

There's no pipe I know of that has black "paint" inside it.

Do the black flecks feel greasy? On the occasions that I've seen this, it's been deteriorating rubber in flexible supply lines, most commonly at the water heater. As mentioned earlier, it ain't paint.

A tub would NOT normally have "flexible hoses" for the water supply. If the specks leave a "skid mark" when you rub them, and if they have a water softener, it may be a reaction of the heater's anode rod with the water.