Oh boy, to think this is good satire is to put a capital letter in Puerile...even though you clearly are a skilled writer.
Most good satire relies on scholastic integrity.
IOW, Jonathan Swift fully understood British bureaucracy, the elites who ran it and the overall goals of the Empire to starve to death a troublesome people...like the Irish during the Potato Famine.
That's why his 'A Modest Proposal' was perhaps 'the template' for good satire.
Peterson spent much of his academic career writing about the rise of fascism, he taught at Harvard, taught at U of Toronto, guided many PHD dissertations, and successfully practiced therapy in Canada.
He knows about thought control.
He's also a Jungian.
Just in case you thought you'd ride in on a tired icon of rabid and vapid anti-intellectualism and post it as satire, here's a tidbit: Jungians often see 'people' who arise to prominence as archetypal manifestations on both the positive and negative spectrum.
Ergo, when a people are downtrodden and labouring under totalitarian dictates, and logic let alone reason are tossed aside like litter from a speeding car of political correctness, an archetype will arise to try and put the brakes on such moronic recklessness.
The point?
Peterson wrote his 'Twelve Rules' because enabling parents, and the institutions promising to educate their children, failed to allow a whole generation to be able to risk failure, to be able to withstand criticism and engage in reasoned debate.
Most academics understand that 'feelings' have no place in scholastic integrity. And yet, because Peterson dared to question a rising and shrill pseudo orthodoxy like 'Wokeness', he is pilloried with Puritanical zeal.
In short, whole generations have almost been robbed of their resilience. Here, let Mommy talk to that nasty Dean. How dare he not refer to you by your first or last name and not your pronouns...how dare that nasty Professor give you a C-Minus, you tried your hardest, he must be Transphobic or racist or fascist or all three!
The point is that he should not have had to write the damn book in the first place! For God's sake, it's a primer for success and something most should have learned as children.
The fact that Twelve Steps became a best seller is more than proof of my earlier assertion that we have failed to prepare many of our children for the rigours of life.
I'd say your considerable skills would better serve the need for satire if you'd go after those who would aim at the speaker while ignoring the message.