Legal stuff
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You have no rights and can’t get any unless rights are terminated and you’re able to legally adopt them. The best you can do is have a letter written and signed by both parents saying you can take them to doctors and stuff. At least that’s been my experience.
Honestly every state is different and it depends on the individual case circumstance.
Tbh you only have the “rights” that the parent gives you at their house.
We don’t have rights. We have access. & that access is contingent on the relationship we have with the parent. The relationship ends, so does the access to the kids. Etc.
You could look at getting a power of attorney or get paperwork making you a legal guardian. Having you as a legal guardian would just add to their parent rights unless specified otherwise. Plenty of people have to do this for grandparents who are primary caretakers during the day for parents.
Anything medical or legal, you can be given permission with documentation. Often it’s per dr so your partner would sign you as an authorized person on their behalf at the dentist and again at the pcp but it could be different in your state. But your partner could probably just call and find out.