I see it this way:
To define “parallel harmony”, it think it would be good to analise the term a little.
Roughly speaking, harmony takes place when we play notes simultaneously, usually more than two. (That doesn’t necesarily mean that there is a key — although it may suggest it). When we have more than two of this combination of notes (called chords), we perceive the moving of the notes as “voices”: moving melodic lines.
On the other hand, there are three basic forms of contrapuntual motion: oblique, contrary and similar. Parallel motion is a type of similar motion (or at least that’s a way to look at it). In similar motion, the two (or more) voices move in the same direction, but not necessarily result in the same interval as they started. In parallel motion, the two (or more) voices move in the same direction AND result in the same interval as they started. For example, if you have a set of notes C, E and G —in the same octave— and you move them up a minor third by parallel motion, you will get Eb, G and Bb.
So, when we talk about parallel harmony, we mean that the voices that constitute the harmony move in parallel motion.
If you search Parallel Harmony in Google, you’ll find several visual examples of that.
The use of parallel harmony often results in “shifting to "unrelated" keys that are identical”, but it isn’t just that, so I wanted to make sure you know this. I hope it worked.
Luis Hernández mentioned Debussy and I think it will be illistrative to look at his music. The first bars of this prelude by him use parallel harmony.
Notice that in this piece not every possibility of parallel harmony is used. There are many cool things you can do with it!
I hope it helps 🤔😁
P.S.: Also notice that some parallel movement in a piece of music doesn’t necessarily imply that the music has parallel harmony. I was listening to Corcovado by Darius Milhaud, and it has a bit of parallel motion in the harmonization of a melody, but that doesn’t mean that he is really using parallel harmony.