Parallel motion; monophonic?

To be brief: would instruments playing a melody in parallel motion, and starting from different notes be considered monophonic? Or another kind of texture?

I have seen conflicting information online, but mostly enough vaguely to not definitively exclude it.

To me it makes sense to call it monophonic as what seemingly defines a melody is the rhythmic and intervallic relationship between the notes in a passage, and the other melodic lines maintain this same relationship only transposed.

Thoughts?