Once you get the hose unscrewed from the spigot and run it a while, try running some into a clean glass or pitcher. See how cloudy/soapy it looks "fresh out of the spigot".
Then let it sit undisturbed for a few hours, and see if the cloudiness "rises" and leaves clearer water behind. If so, it was probably dissolved gasses ... but from what?
If black specks settle down, the speed with which they settle might tell you whether they are silt-sized or larger.
Sending a sample of "spigot water" plus a sample of "indoor sink water" to the water people ought to get some reply from them like "rust" or "chlorine" or "bacterial buildup".
P.S. If that hose has been on that spigot for years, be careful unscrewing it. It might be possible to rip the spigot right off the pipe, if the pipes are rusting out. (And then your parents might say it was your fault!)
P.P.S.
At first I thought calling the water district or public utilities would be free, and they might do something for you. But what if your laws say that it is the homeowner's responsibility to maintain their own pipes so they don't pollute the town water? What if they said your parents "have to" make repairs?
Still, if there is a danger that a rapidly rusting water line is going to burst and spray water all through your house, repairing the pipe may be cheaper than replacing the house.
I hope you keep us updated as you experiment.
