Peter Benjamin Parker was as much of an Avenger as Iron Man. It took no time at all for the young teen to become a part of the Avenger family.
To Tony he was a surrogate son, young and new to the playing field. Iron man felt that as the only father figure Peter probably had, it was his job to teach and protect him.
Natasha saw Peter for what he was, a strong kid who needed the independence to find his own way. She would be there when he needed, and of course she'd offer him guidance and training; but his path was his own to find.
Cap thought of Peter as a little brother, striving to be there for him just like Bucky had once been. He was fiercly protective, and trued to keep Peter out of danger as much as possible.
As for Hawkeye, he was a father of three. Never ever would he wish his life on one of them, thus he slightly resented the fact that a young teenager was putting his life at risk for a world who might never appreciate it. As time went on though, Clint came to appreciate the teen's abilities, settling on the fact that if he couldn't stop him, he'd have to help him.
Thor viewed Peter much in the same way he viewed Loki, a younger brother. Unlike many of the others, he didn't try to mother hen Peter. He thought of Spiderman as a warrior in training, who needed to fight his own battles as much as possible. He would however, be at Peter's back whenever requested.
Bruce Banner meanwhile, was somewhere in between. Yes, he thought Peter needed to be protected, but also respected that the kid needed to be free to some extent. It wasn't until they started talking science that Bruce forgot he was talking to a teenager. Bruce saw untapped potential in Peter, and often viewed him as a fellow scientist, conversing with him in many realms of sciences.
As for Loki, the trickster thought of Peter as an equal. If Loki had been a Midgardian, he calculated that he and Peter would likely have been the same age because Asgardians age more slowly than Midgardians. To the great disdain of the others, hee and Peter would pick a random day of the week to go on a pranking spree. Meanwhile, on the battle field, Peter and Loki were a formidable fighting team when together.
In an ironic twist, Peter was the only other person besides Thor that Loki called brother. And why not, because In Loki's eyes, that's exactly what Peter was, a brother.
Aunt May had been aware of Peter's alter ego for two months now, and trusted the Avengers to keep him safe. She certainly didn't like what he did, but she knew he had to do it. A lot of tears were shed when Peter admitted that Uncle Ben's death was his fault. They sat and cried together, both in mourning.
While she was horrified at first, in the end, she forgave him, proud to see how far he'd come from that point. It was because of this, that she didn't have the heart to stop him, knowing that this was why he worked so hard as Spiderman. For Uncle Ben.
She came to know the Avengers, and actually became a bit of a mother figure to them.
They were different in so many ways, but one thing bonded them together, their honorary little brother. I mean, he was one of a kind... Right?