House Democrats asked the federal labor board to withdraw a proposed “joint employment” regulation, further complicating the controversial effort to reverse a key Obama administration policy.

The joint employment question determines which business entities are responsible for alleged unfair labor practices in the workplace and for collective bargaining with unions. The issue is crucial for franchises and the many American businesses with a workforce that includes contracted or outsourced workers.

The lawmakers said a recent court decision blocks the National Labor Relations Board from moving to significantly limit the situations in which businesses may be considered joint employers.

A federal...