The way I read it, it's not about asking permission, per se, it's about setting expectations. "Head's up, I'm about to disagree with you." Also by giving them this ability to "opt in" to the disagreement, you're creating a relationship based on trust. In other words, they're hearing you respectfully express your thoughts and feelings, and that makes them _more_ likely to listen to you in the future, versus just barging in with your thoughts unannounced. (Contrariwise, if they say "no" too many times, then they don't trust you or your opinions -- and that's information for you to do what you wish!)
And sometimes? Your boss doesn't want you to disagree, they just want you to execute, saving both of you the time and energy of arguing.
You may still disagree, and you may not like their decision to just execute, but at the end of the day, they're the boss and you aren't. You're free to get another job!
As a leader in my company, I welcome disagreement early on. But at some point, we have to execute and we can't get stuck in a stalemate because I want A and you want B. Someone has to make a decision, and as the superior, that responsibility lies with me (or in some contexts, it lies with my superior, even if I disagree!).
(P.S. You are allowed to disagree with your parents, too!)
It also depends on what you're disagreeing about and whether your manager has enough of a pull to get it changed.
Usually rank and file employees are barred from expressing opinions about strategy at all. Permission or not, you will get ignored. The best you may get is more work as an experiment, with little reward and low chances of success.
I accidentally stumbled on the permission to disagree when arguing with my partner. We both do it now. Sometimes people aren't in the mood to argue/debate. When you ask permission, if they say yes, they usually keep their emotions in check. If they say no, you save the battle for another day.
It's a great way to build respect.
Instead of brown nosing, just disagree.
If you ask permission to disagree, you don't tend to get respect. You become seen as a fucking doormat.
Just speak plainly and clearly.
Me, speaking plainly and clearly, Yes
Child: is the macaroni statue I made of you pretty?
Me, speaking plainly and clearly, no, it's hoorendous
Me: Only when compared to this stunning macaroni statue.
I prefer to dodge answers and let others argue it out
Yes you look fat, and incredibly sexy.
No, it’s not pretty, but I love that you’ve made a statue of me.
There are quite a few good ideas in tfa. The power deference is frustrating, but it's also a reality with groups of humans. In the end, you can build up your own forms of informal power and social capital if you deal with this kind of situation pragmatically. As a result you can become someone people will trust to discuss ideas effectively.