They can. TD status for a spouse or an unmarried child under twenty-one rests on proving the relationship, and a marriage or birth certificate showing an earlier surname needs the same linking documents. Each family member's own records should be reviewed individually.
Check the relationship proof against current names
A marriage certificate often does double duty here, proving both the relationship and the name change. A child's birth certificate may show a parent under a former surname, which is ordinary but should be anticipated rather than discovered late. Steps differ by nationality, since Mexican citizens generally require a visa and Canadian citizens generally do not, so confirm what each dependent needs instead of assuming the principal's route applies to everyone.
Dependants' records age differently from the principal's. A child's birth certificate is fixed once issued, whereas a spouse's identity record may change again between a first application and a later extension, so the chain assembled today can need one more link the next time status is sought. Note the date of every relationship document alongside the principal's, and check before each subsequent filing whether anything has moved rather than reusing an earlier folder untouched.
TD status permits study but does not authorize employment, so a spouse who intends to work needs an independent basis, assessed on its own timeline and not folded into the principal's.