No. A spouse and unmarried children under twenty-one may accompany him in TD status, and TD dependants may study, but TD carries no work authorization of any kind.
TD is a permission to live and to study, and nothing more
Derivative status follows the principal, and TD is one of the more restrictive derivatives. A TD spouse cannot take a job, cannot freelance, and cannot run a business from inside the United States, including remote work for a Canadian employer. Children in TD may enrol in school.
Two limits deserve early attention. A spouse must be a legally married spouse; a common-law partner does not qualify for TD and would need an independent basis to come. A child ceases to qualify at twenty-one, which matters to any household with a nineteen or twenty-year-old, because the alternative route for that person takes months to arrange.
Where a spouse's own career is central to the decision, it is worth comparing TN with categories whose spouses are employment-authorized incident to status, such as L-2 and E-2, before the household commits.