No. TD eligibility rests on the family relationship and the principal's TN status, not on how the employer administers payroll. A spouse and unmarried children under twenty-one may qualify, with steps that vary by nationality, and each person's own documents matter.
Separate benefits enrolment from immigration status
Families sometimes read a benefits package covering dependants as confirmation of their immigration situation. It is not; health coverage administered by a payroll organisation says nothing about status. Prepare each family member's documents on their own terms, remembering that TD permits study but does not authorize employment, including remote work, and that a child who is a United States citizen does not need TD.
Two clocks run here and they are not synchronised. A dependant's authorized period is tied to the principal's, so a period granted for less than requested carries the family with it, while a school year, a lease and a benefits enrolment period each run on their own calendar. Decide in advance how the household responds to a period shorter than the one requested.
Each dependant's admission record should be read on the day it is issued rather than filed unopened, since the date recorded there governs whatever the enrolment portal displays. A spouse who intends to work needs a separate basis for it, and qualifying for one is its own process on its own schedule, which the principal's filing does not begin.