Okay, I think my understanding of electromagnetism is a bit off, because I don't completely understand what happens here:

Say we have two positive charges moving parallel to each other at the same velocity relative to me (the observer). I'll observe a magnetic field around each charge's path which attracts the other charge, so the charges come closer together. However, from the charges' frame of reference, since they are stationary relative to each other, won't they repel because they are the same charge?

You have correctly identified a paradox in classical electromagnetism. It's really this, more than anything else, that led to the developement of special relativity. If you model this using special relativity, the paradox does not exist.

Basically, the repulsive force will always be stronger than the attractive one, for speeds less than c. So seen from the stationary frame, the charges will always be repulsed, byt the repulsion will be weaker the faster they move.

However, in the co-moving reference frame, only the repulsive force exists. However, due to the length contraction along the axis of movement due to relativistic effects, even though the charges will move apart faster, they will still follow the exact same path through space as when observed from the stationary frame.

You can also consider the measurement of the separation of the two charges at set points in time, and how time dilation will affect this measurement.

So if you are an electron, you'll see the others move away due to repulsion quite fast, but this happens very very slowly if you look at it from the outside I take it. (I'm talking about free electrons, not your a/c line charges jumping around valance orbits of copper wire since I assume they move quite slowly.)