Yeah that's not what parallel means. Parallel doesn't describe intervals, it describes harmony. If you have a melodic line, and then you write harmony for that line where every note is exactly the same interval from the melody, that's parallel. And it doesn't matter what the interval is; it could be perfect fifths, minor thirds, augmented fourths (i.e. the tritone), etc.
Here's an example. I've got a C major scale playing, and for the harmony, I have parallel major thirds playing on top of it.
E F# G# A B C# D# E C D E F G A B C
Notice how we basically ignore the fact that C major doesn't have any sharp notes in its key when writing parallel harmony, because we're specifically writing a major third up from every note.
On the flip side here's what we call "diatonic" harmony. Diatonic refers to staying within the key.
E F G A B C D E C D E F G A B C
Notice how as we go up the C scale, the thirds on top switch back and forth between major thirds (C-E, F-A, G- and minor thirds (D-F, E-G, A-C, B-D). The harmony on top is staying within the C Major key. This is the kind of harmony you usually hear in songs.