Finland advanced earlier in the day, eliminating Russia with an 86-83 win in double overtime. Russia was the bronze medalist at the London Olympics last year and at the Europeans two years ago. The top three teams from each group advance.
Italy stayed perfect after four games with an 81-72 victory over previously unbeaten Greece, with Chicago Bulls guard Marco Belinelli scoring 23 points. Both teams joined Finland in advancing from that group.
In Jesenice, Serbia beat Latvia 80-71 behind 25 points from center Nenad Krstic to advance.
Ukraine stayed with France until Parker took over in the fourth quarter. The Spurs guard connected on two jumpers and then sank a 3-pointer to help the French pull away. Parker scored 13 of his team's last 17 points. France was runner-up two years ago.
"We kept fighting and in the fourth quarter we found the solutions," Parker said after playing his 50th game in a European championship. "I try to lead the team and to help the young players. When you are playing must-win games, you have to be disciplined, serious. We'll raise our level in the second round."
Five players scored in double figures for Spain, which led 49-13 at halftime and by as many as 48 points before slowing down in the second half.
Krstic said Serbia dedicated its victory to 18-year-old Serbian women's basketball player Natasa Kovacevic, who had a leg amputated after a bus carrying her Hungarian team crashed Saturday.
"We are all affected by this tragedy and she was in our thoughts tonight," Krstic said.
In other games, Bosnia-Herzegovina beat Macedonia 62-54 in Jesenice and Britain eliminated Germany with an 81-73 victory in Ljubljana.