Giampiero Ventura's men were on course to secure sixth spot when they led Parma, who are also in the hunt for European qualification, courtesy of Ciro Immobile's first-half strike at Stadio Olimpico last Sunday.
Parma had other ideas, though, and Jonathan Biabiany was on hand to equalise 19 minutes from time as he followed up after Antonio Cassano failed to score from the penalty spot.
Three points would have ensured the Turin outfit took the final spot up for grabs for Italian clubs in next season's Europa League, so the pressure will be on when they face a Fiorentina side already assured of their place in the competition this Sunday.
Torino's cause was not helped by Serie A top scorer Immobile's red card for a second bookable offence against Parma.
The striker, who has scored 22 top-flight goals this season, will sit out Sunday's crunch clash due to suspension and will be sorely missed.
Immobile's spirits were raised this week when he was named in Italy's provisional 30-man squad for the FIFA World Cup along with team-mates Matteo Darmian and Alessio Cerci.
Ventura's charges have hit form at the right time, reeling off seven games without defeat and dropping only four points in the process but they will face a stern test at Stadio Artemio Franchi.
Biabiany's strike ensured Parma remain in contention for a place in Europe and they must fancy their chances of thwarting Torino ahead of their home clash with relegated Livorno, whose fate was sealed when they suffered a 1-0 defeat against Fiorentina last weekend.
The return of Davide Nicola failed to have the desire effect as Livorno were condemned to the drop and they have picked up just one point from their last nine matches in a dismal end to the campaign.
Milan are also in contention to finish a disappointing season on a high note by taking sixth spot, as they are just two points behind Torino ahead of their clash with in-form Sassuolo at San Siro on Sunday.
Verona are still in with a shout of taking the final European spot as they level with Milan on 54 points, but they have an inferior goal difference and face a difficult trip to third-placed Napoli on the final day.
Champions Juventus only need a draw at home to Cagliari to take their record points tally to a century, while second-placed Roma travel to Genoa.
Bologna and Catania were also consigned to the drop last weekend and they will attempt to sign off with victories over Lazio and Atalanta respectively.
Elsewhere, Inter are also on the road at Chievo and Udinese host Sampdoria on Saturday.
