Well, as with a lot of things in tango, the word "parallel" is wide open to interpretation, and mostly means "the way i like it" (or, in some cases "the way i don't like it").
That said, yes, one can dance literally parallel, though to make this a satisfying dance requires (imo) a specifc technical framework that facilitates free and energetic movement in that space
1) both the leader and follower need a strong forward+upward presence - the follower is moving diagonally up through the leader, the leader is moving diagonally up through the follower
2) both move by stabilizing their core and moving the legs, but this is different than the ballet conceptualization of the same words (ballet stabilizes the pelvis, and moves by spiralling within the thigh - this technique frees the pelvis, stabilizes the spine, and opens and closes the pelvis by using the diagonal muscles of the back)
3) the pattern of movement and relaxation is reversed - the couple is most energized when standing still and the forward presence creates a coiled energy, and movement is about relaxing the resistance to that coiled energy , i.e. the leader does not "push", but the follower relaxes her resistance and decides to carry the energy somewhere else (this is a interesting concept to play with - and a lot of traditional tango vocabulary makes a lot more sense when powered this way).
The easiest way to figure out if a follower enjoys dancing this way is to start walking in the cross system on the right side (from the leaders perspective). If this works smoothly and powerfully using this framework will work for the whole dance. If not it is much better to switch to a different technical framework.