I just bathed my dog with sergeants skip-flea & tick shampoo with 0.10% permethrin and 0.50% piperonyl butoxide, technical. My dad then proceeded to almost immediately after apply another flea treatment to her neck, ingredients or brand unknown to me as he refuses to let me see it since he “knows what he is doing”. Essentially, I am simply worried about my dog and want to make sure that my father hasn’t potentially caused her future death or harm.
I am not an expert, but to calm your fears, I do not believe there will be an issue.
I located the EPA hazards notice for the product I did not find a warning there, nor did I find one at several other places I looked.
You should watch for pet for any unusual signs/symptoms and consult a vet if needed.
It will have been around 24 hours since the application as I write this answer. If there have not been issues yet I doubt there will be any problems.