Celtic's one-season stay in the top flight of non-league football ended with a fifth successive loss combined with wins for Halifax and Northwich which pushed the possibility of a last-day escape beyond Deacey's men.
* Click here for a picture slideshow of match action from the game.But the man who stepped up to replace Lee Sinnott as manager last November hopes Saturday's game against Grays Athletic will not be his last as Celtic boss.
Deacey said: "I want to stay with Farsley Celtic but it's up to the board to see what they want to do."
Deacey looked as though he would maintain Celtic's premier status earlier this month when back-to-back wins lifted the club to 18th in the table.
But failure to take a point since proved costly and despite a brave effort by his side at York, when it looked at one stage as though they might force a draw to keep the dream alive, Deacey was unable to stop the rot.
He added: "I am gutted. The whole dressing room is gutted but we have got to bounce back. It's character building. We had the success last year and we've got to learn. We haven't disgraced ourselves in this league. We have given it our all, especially in the second half last night, but it just hasn't worked out this year.
"We have had a lot of complications but I feel we've given a good account of ourselves and the players should be proud, along with the committee and the supporters."
Two goals in the opening 20 minutes at KitKat Crescent sounded the death knell for Celtic as York went on to end their home campaign on a high.
The ever-dangerous Martyn Woolford put City ahead when he collected a flick-on from Leo Fortune-West and despite desperate appeals for off-side was allowed to continue and poke the ball under Curtis Aspden to claim his 17th goal of the season.
The Farsley keeper then made a superb save to deny Woolford when he touched his header round the post but that merely delayed the punishment for few seconds as Fortune-West volleyed in from the corner.
To their credit Farsley didn't fold and after David McNiven went close to pulling one back just before the break they launched a second-half fightback.
Sub Rory Prendergast gave Celtic an extra cutting edge, particularly from set-pieces, and from his deliveries Amjad Iqbal and Simeon Bambrook went close before Bambrook headed in a corner.
That gave Celtic 16 minutes to save themselves but after Bambrook headed another Prendergast free-kick over, York exploited defensive gaps to add two late goals through Richard Brodie and secure their biggest home league win of the season.
YORK CITY: Evans; Purkiss, Parslow, McGurk, McWilliams; Rusk, Beadle, Panther, Woolford (Brodie 71); Sodje (Shepherd 82), Fortune-West. Subs not used: Robinson, Boyes, Mimms.
FARSLEY CELTIC: Aspden; Downes, Sambrook, Iqbal, S McNiven; Watson (Prendergast ht), Knowles, Krief; Reeves, Torpey, D McNiven (Heath 67). Subs not used: Stamer, Tuck, Backhouse.
REFEREE: Gary Sutton
ATTENDANCE: 1,886.
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