The First Amendment does not prevent a state from regulating treatment even when that treatment is performed through speech alone.
Scientifically discredited and dangerous practices are not safeguarded by the First Amendment (e.g., doctors are sanctioned when they express false or harmful medical opinions to a patients). And to be clear, conversion therapy can be extremely dangerous and, in some cases, fatal. Conversion' therapy, is at its core based on an understanding of sexuality that has been summarily rejected by virtually all the major health and mental health professions. The practice is opposed by the American Medical Association, the American Psychiatric Association, the American Academy of Pediatrics, and every other mainstream medical organization in the United States
While some would argue parents have the constitutional right to control the upbringing of their children, this right does not allow them to subject children to harmful forms of treatment. Banning conversion therapy fits squarely within a state's well-established authority to prevent health care professionals, including therapists, from harming their patients.