Non-sequitur, some variation of Red Herring, or...?

“Your parents had legal custody of you as a child and therefore by choosing to birth you in the United States, they opted you into the social contract. If you disagree with that, then you must disagree with the concept of parental custody, in which case you would be arguing that 3-year-old's should be legally independent of their parents…”

As others have pointed out, the part in bold is a false premise. Custody refers to parental responsibilities, not child responsibilities. There's no 'social contract' for children, therefore, disagreeing with the idea that you're opted into it does not mean you disagree with parental custody - until the age of majority or such a time where the child is able to take care of themselves (and thus, is no longer a 'child or 'minor' for legal reasons) the parents are still legally bound to that person.

Pakman is somewhat conflating 'social contract' with 'custody' and then applying it to the wrong element (if they're supposed to be different concepts).

So it sounds like a non-sequitur.

The ultimate issue here appears to be that Pakman is setting requirements for legal custody and making it binary; either something someone accepts or doesn't. Assuming he is factually incorrect (a lawyer who specializes in this area would know), he does appear to be committing a False Dilemma .

Aside from that observation, Pakman's statement, as it is posted, is confusing and appears meandering. This is certainly a case where I would ask for more clarification.

"Child custody consists of legal custody, which is the right to make decisions about the child, and physical custody, which is the right and duty to house, provide and care for the child"

Parents make decisions on behalf of the child. No fallacy. 

Isn't this simply a case of mixing terms with ambiguous meanings? Inheritance is biological concept, with legal meanings as well, but ideas are not inherited or inheritable.   The social contract is an idea. It must be taught.

Custody is a responsibility of the parents, but imo comes attached with authority to act on behalf of the child.  In the US, laws differ on what age is sufficient for a child's assertion of emancipation, but none agree with 3 afaik, or even with 14 (when some marriages are allowed, but the child is still not entitled to vote, run for office).

With no dog in this hunt (all my children are married and independent) I say the rights and responsibilities are linked.

We teach (may I so presume) the social contract.  We hope it's subscribed, but even Bonnie and Clyde had parents, and the parents were not culpable imo for their failure to understand the values taught. Horses to water...

To get to the nitty gritty and directly attack the one of the ideas, the custody enables a responsibility to try to imbue the social contract but does not convey a responsibility for success of the child to incorporate the teaching.  As I see it, the contract expects one to try, to teach. but there are no guarantees.

If you disagree with that, then you must disagree with the concept of parental custody,

If an argument is to succeed it must disconnect the rights and responsibilities.  There is no inheritance of an idea.