Can a child therapist work only with little kids or with the whole family?

Most Kid +. Most of the therapists i work with include family/parent interviews as a way to understand family dynamics. These interviews may be conducted with or separate from the kid. As treatment evolves, the focus is on the kid but may be modified to include others as treatment progresses. No "one size fits all" approach will work with the diverse issues involved.
Usually family too. Child psychoanalysts ie anna freud just worked with kid but all others that i know involve at least the parents.