Poleon did give Latics the lead on nine minutes but Bostwick's goal 15 minutes from time sealed a final day draw for Dean Holden's men.
Latics started the game brightly but it was Posh who had the first shot of the game on two minutes but it went harmlessly past Joel Coleman's post.
Mike Jones' set pieces were causing an early problem for Peterborough's defence and Danny Philliskirk nearly capitalised on a free kick but Ben Alnwick saved well.
Athletic did take the lead on nine minutes through Poleon.
Brian Wilson's pinpoint cross found the head of the number nine and he notched his seventh goal of the season in all competitions to put Latics ahead.
Latics continued to press their opponents but there was a break in play on 16 minutes after a nasty injury to Erhun Oztumer.
He was replaced by Jack Payne on 20 minutes but seconds after play resumed, Latics almost doubled their lead.
Rhys Turner knocked the ball past the defence, raced through on goal but Alnwick saved well at his far post.
Joel Coleman, who was the recipient of the April Player of the Month award, was being put under increasing pressure by Posh but he did well to punch away corner after corner.
Poleon was proving to be constant threat on the left-hand side and just after the half hour mark he went close to making it two inside the penalty box but Alnwick tipped wide.
Posh's Joe Newell was a bright spark for the visitors as he flashed a shot past Coleman's goal five minutes before the break.
Peterborough were forced into their second change of the afternoon on 40 minutes as Gabriel Zakuani limped off with what looked to be a hamstring injury to be replaced by Shaun Brisley.
Conor Washington could have levelled for Peterborough into added on time in the first half but Coleman recovered to save well.
The visitors came back into the game in the second half as Newell again was thwarted by Coleman from 30 yards.
The game seemed to lack energy but Jordan Bove's introduction seemed to spark Latics into life.
The 19-year-old flicked the ball into the path of Philliskirk and almost received it back from the forward but the Posh defence just recovered to clear the danger.
Latics were on top and central defender James Wilson was the next to go close but his volley was saved well by the Posh goalkeeper.
Peterborough were reduced to ten men on 67 minutes as Edwards lunged in late on Wilson and he received his marching orders from the referee.
Dean Holden brought on Jack Tuohy for his first professional Football League appearance on 68 minutes replacing Carl Winchester.
The ten men of Peterborough seemed to be lifted by their one man disadvantage and almost equalised 15 minutes from time.
Conor Washington managed to get behind the defence but J.Wilson managed to get back to block the ball and it rebounded out for a corner.
From that resulting corner, Posh drew level through Bostwick. Latics failed to deal with the set piece into the box and the midfielder fired him from inside the box to equalise.
Tuohy didn't seem overawed by the opposition and almost played Connor Brown through on goal but the defender couldn't take the ball in his path and the chance passed.
Latics could not muster a goal to win the game and finish the 2014/15 season in 15th position.
Oldham Athletic: Coleman, B.Wilson (Dieng 87'), J.Wilson, Gerrard, Brown, Kelly, Jones, Winchester (Tuohy 68'), Poleon, Philliskirk, Turner (Bove 60').
Subs not used: Kean, Elokobi, Woodland, Mellis.
Peterborough United: Alnwick, Smith, Bostwick, Oztumer (Payne 19'), Washington, Newell, Santos, Zakuani (Brisley 39'), Anderson, Edwards, Lopes (Maddison 71').
Subs not used: Henry, James, Burgess, Beautyman.
Goals: Oldham Athletic (Poleon 9') 1-1 Peterborough United (Bostwick 76')
Attendance: 3,943 (324 visiting supporters)