With the win, France is also back in the Olympics for the first time in 12 years.
"I've been chasing this for 11 years and I'm very happy that I can finally say I'm going to the Olympics," Parker said. "I had tears in my eyes when the buzzer came and I knew we'd won the game."
Juan Carlos Navarro scored 35 points in a performance his coach called a "masterpiece" as Spain beat Macedonia 92-80 in the first semifinal on Friday.
The European championship finalists automatically qualify for next year's London Games.
In its most recent appearance at the Olympics, France won the silver medal at the 2000 Sydney Games and Parker led the country's Under-18 team to the European title.
France, which also got 19 points from Nicolas Batum, reached the championship game for the first time. It won silver in 1949, but the tournament was played in a different format and did not have a final.
France won bronze medals in 1995 and 2005.
"It was our year and we played great tonight," Parker said.
Parker was the MVP of that junior team in 2000. He has gone on to win three NBA titles with the San Antonio Spurs, including being the MVP of the 2007 finals, and has been selected to three NBA All-Star games.
Tied at 45, France went on a 10-2 run to finish the third quarter for a 55-47 lead.
The lead stayed in double digits into the fourth when Ali Traore scored to make it 61-49. He had eight points in that key run.
"We know Ali is a huge potential offensively and we use him to score," France coach Vincent Collet said.
"Our goal was to qualify for the Olympics and we played our best game. We controlled all the best assets of the Russian team and the rebounding, especially Batum."
Batum's seven rebounds helped France control the boards against Russia, which was undefeated until the semifinal.
"Tony was Tony, but the guy who really hurt us was Batum, he made big plays at big times," Russia's American coach David Blatt said.
Batum, who plays for the NBA's Portland Trail Blazers, had a three-point play for a nine-point late in the fourth. He also had a drive for a dunk for a 75-66 lead and grabbed a big offensive rebound with less than a minute remaining.
Andrei Kirilenko led Russia with 21 points.
Navarro scored 19 points in the third quarter to break open the game. Spain coach Sergio Scariolo did not hold back in praising his point guard.
"He's like a masterpiece you see in Louvre or Prado, he is like one of the best artists, he has his own inspiration ... He is out of this planet," Scariolo said.
"We are very proud to have reached the goal of qualifying for the Olympics. It was tough. They play with heart and class and they play smart."
Macedonia has been the surprise team of the tournament and led 45-44 at halftime.
Then Navarro took over.
"He was amazing, incredible, he did everything on both ends," said Spain power forward Serge Ibaka, who had 11 points.
Ibaka, a native of Congo who plays for the Oklahoma City Thunder of the NBA, is in his first tournament with Spain, his adopted country.
Macedonia coach Marin Dokuzovski said his team came into the game "as always with a big heart and to play hard."
"We knew whom we were playing and that it would be hard to surprise Spain. We didn't play such good defense as before, my players were getting tired and Spain's offensive rebounds hurt us," Dokuzovski said.
Bo McCalebb, the American-born Macedonia point guard, led his team with 25 points and five assists, while forward Pero Antic had 17 points and nine rebounds.
Pau Gasol, who struggled with 8 of 20 shooting, had nine offensive rebounds as Spain outscored Macedonia 22-6 in second-chance opportunities.
Younger brother Marc Gasol had 10 rebounds and 11 points.