City are winless in four league matches, while the visitors enter the game on the back of successive wins to open up a nine-point gap on their opponents.
One point collected from the last 12 on offer has seen City slide out of the playoffs and the same old questions asked of boss Lee Johnson.
The Robins have been unable to see through promotion challenges in recent times, though they did at least end a three-game losing run in last week's 1-1 draw at Millwall.
As it is, with just 10 games of the season to go, they find themselves two points behind Preston North End in sixth.
Of those 10 remaining matches, three games are against sides above them, including Saturday's visit of high-flying Fulham.
The Cottagers are providing the biggest threat to West Bromwich Albion and Leeds United for automatic promotion, but the gap on the two sides above them remains at six and five points respectively.
They have been held 1-1 in back-to-back away games, though, and have won on their travels just twice in nine attempts in all competitions.
However, Fulham are unbeaten in their last four away league games against Bristol City since a 5-1 defeat in April 1990, winning three of those matches.
Another win at Ashton Gate this weekend will see them all but seal a place in the end-of-season playoffs, though Parker will no doubt still have his sights set on breaking into the top two.
Bristol City's Championship form: LWLLLD
Fulham's Championship form: WDLDWW
Famara Diedhiou has scored three times in his last seven outings, meanwhile, and is expected to retain his place upfront.
Aboubakar Kamara boasts the best minute-per-goal ratio of any player in the division to have played at least 500 minutes, and his form could see him rewarded with a start here.
If that is the case, Anthony Knockaert is the favourite to drop down to the bench after failing to play a part in a goal in Fulham's last four matches.
Bristol City possible starting lineup:
Maenpaa; Pereira, Kalas, Baker, Dasilva; Paterson, Smith, Nagy, Eliasson; Diedhiou, Wells
Fulham possible starting lineup:
Rodak; Christie, Hector, Ream, Odoi; Arter, Cairney; Kamara, Reid, Cavaleiro; Mitrovic
We say: Bristol City 1-2 Fulham
Since the start of December, only Hull City (six) have lost more home Championship games than Bristol City (five). The Robins picked up a point in their most recent outing away at Millwall, but Fulham have regained some momentum and we are backing them to win this one.