Relative key and parallel key are both terms for the relationship between two keys, but they use different reference points. Relative keys are a major key and a minor key that share the same notes, while parallel keys are a major key and a minor key that share the same tonic. The terms are similar, but you use them differently when writing.
Relative keys: sharing the same notes
C major and A minor are relative keys because both can be made with the white keys alone. Even though the notes of the scale are the same, whether you feel C or A as the center changes the impression of the song and where the chords land.
Parallel keys: sharing the same tonic
C major and C minor are parallel keys because both are felt around C. Their notes, however, are different. When you want to move back and forth between major brightness and minor shading, the parallel key relationship is what helps.
The relationship with borrowed chords
With borrowed chords, borrowing from the parallel key is a very common approach. Putting a few chords from C minor into a song in C major changes the mood without losing much of the sense of key.
The relationship with modulation
If the center of the song moves to the relative or the parallel key, it can be heard as a modulation. On the other hand, if you only use different chords for a few bars, it is often clearer to think of it as borrowed chords.
How to check it in a DAW
Lay the C major and A minor scales out in the piano roll and confirm that they use the same notes. Then compare C major and C minor to see visually that the center is the same but the notes change.
