Did Tom Paris need to graduate from the Academy in order to join Voyager crew and how was he able to do that?
That is tough to answer.
[Paris graduated from SFA, and] was assigned to the USS Exeter. His career in Starfleet was short-lived, however, and ended after he was involved in covering up his own piloting error which had led to the death of three fellow officers at Caldik Prime. Despite later telling the truth, he was discharged from Starfleet following the incident. He once remarked that "the ghosts of those three dead officers came to me in the middle of the night and taught me the true meaning of Christmas" when asked why he finally confessed. (VOY: "Caretaker", "Non Sequitur")
After being expelled from Starfleet, he wandered around "looking for a fight" and found it in the Maquis. He was with them for a few weeks before he was captured on his first mission, convicted of treason, and sentenced to eighteen months' imprisonment at the Federation Penal Settlement in New Zealand.
He was the son of a Starfleet Admiral, Owen Paris, whom Captain Janeway had served under, as well as being a graduate of Starfleet Academy. So she gave Tom Paris a second chance. He was also a part of the Maquis. His inside information was a way for Janeway, in my opionion, to justify asking for him to be released.
"Did Tom Paris need to graduate from the Academy in order to join Voyager crew?"
I would say: Yes, as he would have to know how to pilot a Starfleet Starship to join the crew.
Did Tom Paris need to graduate from the Academy in order for Janeway to ask him to help find the Maquis? I would say: Yes again. In my opinion, she couldn't have been able to justify asking for him just on his association with the Maquis no matter how valuable to Janeway that information may be. Having once been part of Startfleet and being a graduate of SFA was, in my mind, why she wanted to give him (the son of a mentor) a second chance.
Again: All this is my opinion.
I don't believe he needed to have graduated from Star Fleet Academy to get on Voyager during the mission into the Badlands. Janeway recruited him because he knew the area and was being picked up as an observer
Paris reminds Janeway that because he was only with them a few weeks, he didn't know many of their hiding places. Janeway replies that he knows the territory better than anyone in the Federation.
Janeway says that the Federation will help him at his next parole review. Janeway informs him that officially, he'd be a Starfleet observer, to which Paris replies that he's the best pilot Janeway could have. Janeway is firm about Paris' role and once the mission is over he'll be 'cut loose', causing Paris to agree with the rejoinder, "Story of my life."
and had everything gone normally he would have then left. however after they are trapped in the Delta Quadrant Janeway makes the decision to make Paris and the Maquis Crew Starfleet officers which i would think normally wouldn't happen.
the point of the "qualification" of graduating from Star Fleet Academy not being needed after the Cartaker Array was destroyed is shown with the Maquis crew who joined as at the very least out of all of them B'Elanna Torres had dropped out
Her teachers included Commander Zakarian and Professor Chapman. Torres and Chapman argued constantly, so she was later surprised to learn that Chapman thought she was one of the finest students he had ever taught and was disappointed when she dropped out. He had placed a note in her permanent record saying he would support her if she ever reapplied.
and B'Elanna's own rank on Memory Alpha is listed as Lieutenant junior grade which is positioned between an ensign and a full lieutenant while Harry Kim is still listed as Ensign and he graduated from the Academy as valedictorian of his class. so Academy background wasn't really a strict requirement once Voyager was stuck in the Delta Quadrent
Of course, while moot now, Tom did graduate the academy
Paris managed to graduate from Starfleet Academy after a stormy four years with a major in astrophysics. (VOY: "Lifesigns", "Future's End")