Having already seen their hopes of progressing to the Champions League last-16 dashed, the Dutch side are playing for third place in Group F.
Yet they wasted a chance to seal their spot at Parc des Princes, slumping to a loss despite performing impressively.
Indeed, it seemed Ajax might grab a surprise win when Davy Klaassen cancelled out Edinson Cavani's opener midway through the second half.
But Zlatan Ibrahimovic restored PSG's lead before Cavani struck again to leave Ajax facing a make-or-break home clash with APOEL.
"If there was a night to qualify for the Europa League then this was definitely it," said De Boer.
"We got to 1-1, but went back behind due to two errors. We should have won here actually; it's a crying shame."
Goalscorer Klaassen echoed his head coach's sentiments, adding: "We needed to get out more [from the game]. PSG was ripe for the slaughter."
Bizarre Jose Mourinho statement aims to pre-empt punishment after Fenerbahce boss clashes with Galatasaray counterpart
'Andy Robertson has a decision to make as a player, where you either roll your sleeves up and give someone great competition or you accept decisions and say that your time is done.' Alan Shearer on the conundrum facing Arne Slot's Liverpool
