Left-back Parkhouse was repeatedly roasted by Wanderers flyer Jumpei Kusukami early in Sunday's 3-1 capitulation, allowing the hosts enough space on the flank to open the scoring within 10 minutes.
The 24-year-old was substituted for Tom Doyle after just 30 minutes, but Buckingham revealed there was more behind his inability to keep up.
"They were getting a lot of success down their right-hand side through Jumpei and some of the movement of Mitch Nichols, and Adam wasn't dealing with it," he said.
"We couldn't let it continue. Subsequently Adam has under-reported a sickness, a virus, that he felt late last night and this morning and which he told us about after 20 minutes of the game."
Buckingham was highly critical of Parkhouse's failure to inform staff he wasn't well before the match, a result that has all but mathematically ruled the Phoenix out of a finals berth.
"It's a big game for us ... it's not a game that we can carry players," he said.
"When you've got a ready, able replacement in Tom Doyle who could have started the game and had 11 fit, fresh players on the pitch, that's hugely disappointing for the playing group, the staff, and the club.
"We needed every single player to step up this evening, and when you start with effectively 10 men you struggle to do that.
"We'll deal with that when we get back to the hotel and get back home."
The loss, softened only but Gui Finkler's late free-kick, leaves Wellington in seventh, five points behind the sixth-placed Wanderers and up against it to neutralise the deficit with four rounds remaining.
Compounding their misery, the Phoenix will lose six players to international duty for next weekend's home clash with Newcastle.
"They now open up a five-gap lead so it's Western Sydney's to lose," Buckingham said.
"We'll still be trying to push for that sixth spot. It's very much in Western Sydney's hands now."