His goal extended Inter Milan's lead to 2-0 by halftime at Parma but the Serie A front-runners couldn't hold it and ended up drawing 2-2 on Saturday.
Inter inched four points clear of second-placed Napoli, who visit Bologna on Monday.
Parma's fourth straight draw moved them four points above the relegation zone.
Inter are in the hunt for a treble - which they achieved in 2010, under Jose Mourinho. The Nerazzurri drew with AC Milan 1-1 in the first leg of their Italian Cup semi-final midweek and play at Bayern Munich in the Champions League quarter-finals on Tuesday.
Despite the intense schedule, coach Simone Inzaghi opted not to rotate his squad and Inter took the lead in the 15th minute thanks to several strokes of fortune.
Lautaro Martinez's attempt at a one-two was intercepted by a Parma defender but the ricochet came to Federico Dimarco, whose cross was deflected by another Parma player to a completely unmarked Matteo Darmian. His effort clipped the right post but rolled in.
Inter goalkeeper Yann Sommer made two fantastic saves inside the opening 20 minutes before the visitors doubled their lead in even stranger circumstances.
Thuram spectacularly shinned a cross from Henrikh Mkhitaryan but it looped up in the air and goalwards. Pontus Almqvist tried to clear it off the line but the Parma winger kicked it off his own face and in.
Thuram's father, World Cup winner Lilian Thuram, was watching from the stands and both he and his son looked embarrassed.
Thuram was born in Parma while his father was playing for the team.
It was Thuram's 14th league goal this season - 17th across all competitions - and a personal record for the France international.
Parma got back into the match on the hour mark when halftime substitute Adrián Bernabé drilled into the bottom right corner from 20 metres for his first goal in Serie A.
Another substitute helped Parma turn the match around completely nine minutes later when Jacob Ondrejka's effort was deflected past Sommer.
AC Milan also had to fight back from two goals down to draw 2-2 at home to Fiorentina.
The Rossoneri got off to a dreadful start as they found themselves 2-0 down inside 10 minutes. Milan defender Malick Thiaw bundled the ball into his own net in the seventh minute and Moise Kean volleyed in Fiorentina's second three minutes later.
United States international Yunus Musah was at fault for both goals and he was hauled off in the 23rd, moments after his compatriot Christian Pulisic combined with Tammy Abraham for the latter to pull one back for Milan.
Musah's replacement, Luka Jović, scored the equaliser in the 64th.
Milan remained ninth, four points below eighth-placed Fiorentina.
Bottom club Monza look ever more doomed after losing at home to Como 3-1 left them 10 points from safety.
Como moved 10 points above the bottom three.