WASHINGTON—The Federal Bureau of Investigation has set up a process to track threats against school-board members and teachers, moving to implement a Justice Department directive that some law-enforcement officials and Republican lawmakers say could improperly target parents protesting local education policies.

The heads of the FBI’s criminal and counterterrorism divisions instructed agents in an Oct. 20 memo to flag all assessments and investigations into potentially criminal threats, harassment and intimidation of educators with a “threat tag,” which the officials said would allow them to evaluate the scope of the problem.