When Meredith Heller adopted Bodie, a young Beagle mix, she knew shelter dogs often have trouble settling in. But she wasn’t worried. Among the benefits offered by her Manhattan employer, customer data platform provider mParticle: two weeks paid “pawternity leave” for workers adopting a rescue dog.

Ms. Heller and Bodie, who turned out to be perfectly well behaved, spent their time off hanging at the dog run and meeting other canines in their Brooklyn neighborhood.

“It...