No. A spouse or child admitted in TD status may live in the United States and may study, but TD carries no employment authorization and there is no application to obtain it. Restaurant, seasonal or casual work is employment regardless of how few hours it involves or how it is paid.
Give the second adult a route rather than a workaround
Treat the accompanying adult's working life as a separate case from the start. The first question is citizenship, because a partner who is a Canadian citizen may have their own listed profession and could hold TN status in their own right if a US employer offers them prearranged work. The second question is credentials, since some professions in the list have licensing requirements that take months to satisfy in a particular state.
The third is timing, because a dependent who intends to study can begin in TD status but a dependent who intends to work cannot start on the strength of an application that has not yet been made. Where the household cannot make the second income lawful, the sensible response is to rebuild the budget around one income rather than to hope that a few shifts will not be noticed. Hypothetical example: a speech-language pathologist receives a US offer while her partner plans to keep a Canadian remote job.
Their first decision is whether the partner needs independent work authorization rather than TD status. List each person's intended activity, passport nationality, licence needs, and earliest required start date. Then examine a separate principal route before relocation.
The mistake is treating household income as an informal exception: a dependant without work permission cannot lawfully fill a few paid shifts or continue work merely because the employer is located in Canada.