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."
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