The French, beaten by Spain in the 2011 final, will meet Lithuania on Sunday after the Baltic nation defeated Croatia 77-62 on the back of 63 points from three back-court players earlier in the day.
Parker, who has won three NBA titles with the San Antonio Spurs, capped a memorable performance with 32 points as his team rallied from a 14-point half-time deficit to draw level at 65-65 at the end of regular time.
It was heartbreak for Spain as they threw away the chance to win their third European title in a row.
Spain looked home and dry at the interval. France had missed their first 10 attempts from three-point range while Parker scored 14 of his team's 20 first-half points as his colleagues appeared overawed by the occasion.
The tide turned after the break as France hit nine of the next 10 shots from behind the arc with the unstoppable Parker producing blistering lay-ups, lethal perimeter shots and defence-splitting passing having also racked up six rebounds.
With the teams locked together at 67-67 in the closing stages of overtime, Parker and guard Antoine Diot coolly sank four free throws apiece to send their small band of fans into raptures in the Stozice Arena.
Lithuania reached their first final since winning the 2003 edition after playmaker Mantas Kalnietis (18 points), shooting guard Jonas Maciulis (23) and forward Linas Kleiza (22 points and 11 rebounds) overran a Croatian team lacking the shooting accuracy of their rivals.
After a tight first half the Lithuanians engineered an 18-3 run early in the third quarter to take an unassailable 58-40 lead and set up their ninth win in 10 matches against the Croatians.
"We gave Kleiza and Maciulis too many open shots and we didn't play defence with enough tenacity," Croatia's top-scorer Bojan Bogdanovic (15 points) told HRT television.
Coach Jasmin Repesa added: "We hoped to go all the way but looked very flat today and had no chance of winning with such poor perimeter shooting."
Croatia nailed only three of 20 three-pointers.