The Serie A strugglers must secure a big win over Besiktas in their final Group B game to reach the first knockout round.
Parma sacked coach Silvio Baldini on Monday after a weekend defeat by Livorno left them stranded near the foot of Serie A.
The Italian side are still reeling from the financial problems that beset their former parent company Parmalat last year that forced a sell-off of key players.
Twelve qualifying places remain up for grabs from the final round of matches this week.
Newcastle United, semi-finalists last year, can seal top spot in Group D by taking a point at home against Portugal's Sporting, who also need a draw to progress.
The English team, whose domestic form, especially at home, has not matched their UEFA exploits, have a strong incentive to win the group, thus avoiding a Champions League team in the knockout rounds when the competition resumes next year.
The eight group winners are guaranteed a tie in the last 32 against a side finishing third in a group. The runners-up will face one of the eight sides -- including Cup holders Valencia -- who drop in from the Champions League.
Newcastle, who have won all three of their group games to date, will want to take revenge for a pre-season 1-0 home defeat by Sporting. Sochaux need only to avoid defeat against Panionios to also advance.
The battle for top spot in Group E, where the top three teams are already through, should reach an exciting climax.
Middlesbrough, in their first season of European competition, will take that honour if they defeat Partizan Belgrade at the Riverside on Wednesday.
A draw would hand last season's semi-finalists Villarreal the chance to leapfrog both with a home victory over Egaleo.
Ukraine's Dnipro Dnipropetrovsk are already through from Group C with Real Zaragoza, Club Bruges and Austria Vienna seeking the other two places.
A draw at home to Dinamo Zagreb will be enough to put VfB Stuttgart into the next round with Heerenveen also sealing a place from Group G if they defeat Beveren at home.
German second division side Aachen will take the remaining qualifying place from Group H with a win in Greece over pointless AEK Athens. Sevilla and Lille have already qualified.
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