[john70t]

All the water comes from the pipe from the street.

The "hot water" is from side of the split which goes through the water heater. The tank will collect sediment at the bottom.
Noah posted the hot water side is fine, but the cold water is spitting rust.
Just like bleeding the brakes, I'm guessing the rust/silt/sediment on the "cold side" may have flushed through the nearest outlet.

I'd run the cold water until it gets clear again.
May take a little while, but hopefully won't screw up the drainage system.