No. TD status permits study but not employment, and paid tutoring performed in the United States is employment even if the students are Canadian or pay by e-transfer. The spouse needs a classification of their own to work.
Location of the work decides, not the payer
Once a family is in the United States in TD status, the question for any activity is whether it is work performed on U.S. soil. Tutoring students at a kitchen table in the college town is work in the United States, whether the pupils are American or Canadian and however the money arrives.
The TD spouse and children may enrol in school or university, so a spouse who wants to retrain or take courses has a lawful path. A spouse who wants an income has to qualify independently: a college instructor spouse could hold a second TN with a second offer letter, and other listed professions such as accountant, registered nurse or physiotherapist each carry their own credential rule. A child in TD status faces the same restriction and may not take a part-time job.
Plan the household budget on the principal's salary alone until each other person's classification is confirmed rather than assumed.