ESTELLA, Spain—The nationalist parties that rose to power in Spain's Basque Country 1½ years ago promising a peaceful path to self-rule are now finding themselves stuck, even as another northern region moves decisively toward the same goal.

An uneasy peace has prevailed in Spain since the Basque separatist group ETA—Western Europe's last major, homegrown terrorist organization—declared what it called a permanent cease-fire in 2011. More than 800 people were killed during its 43 years of armed struggle for independence.

...