I think Parker is still thinking about his relationship with Walter. Since they seems to be realllyyyy close friends in the past, even more than the other ones. With the whole "Ill protect you no matter what" thing. So yeah, i think Parker is still seeing Walter as a friend, and he is conflicted with himself starting to feel in love with Walter. And of course, he is his student XD.
True, that is one aspect I really enjoyed about Parker. While he definitely having some feelings, his love really seems to be true (not that the others don't), but he so incredibly sincere that it makes me fall for him more! He was also the one to tease Walter, outright requesting that he do something, which goes to show how conflicted he is with his feelings. I also really liked when he gets drunk and immediately starts regretting shooting Walter down and not just being upfront about how he feels. Another piece of fantastic character development! Im honestly a bit traumatized from Juuichi from Morenatstu, whats with these bears and being so secretive about their feelings?! Also Gil doesn't seem to care that Walter is his student xD, go ahead Parker take the plunge!!