Yes, actually it is OK for an Asian person to make fun of themselves.
What a lot of white people miss is that a big factor is relative power (or privilege). If the comedian is an American white guy making fun of black people, there's the whole ugly history of American slavery, oppression, etc. behind it. Whether or not the comedian means to invoke the spectre of lynching, it's something that's more likely to be in the background of how a black person perceives it.
It's not the case that people are suddenly being offended by things, they have always been offended. Just maybe now they're actually feeling like they can speak up about it without fear of brutal retaliation. Imagine you're in a company meeting and the boss, who makes enough to have a Mercedes, starts making fun of your sh*tty old Honda. Are you offended? Almost certainly. Can you say anything back your boss? Probably not.
I'm not sure you are speaking for the entire Asian community, and not all Asians will think the same way you do.
One of my favorite comedians is Louis CK, there is this famous routine in SNL were he makes fun of child molestors, I recommend you to watch it, the balls this guy has to say what he said on National TV are unbelievable
https://www.youtube.com/watch?v=yzh7RtIJKZk, it is the most Un pollitically correct thing I have ever heard, but it is brilliant. Yes, I know, he used to masturbate in front of women and got punished for that by basically making him a pariah. But my point is, that is comedy! being able to make fun of whatever you like, that is also why I love Family Guy, those guys will make fun of everything and everyone, no one is safe, and its hilarious, people need to understand the difference between comedy, free speech, and just plain old hate.
I've said it before, I am Mexican, when I hear a comedian making fun of Mexicans I do not get offended, I laugh, because most of what they are saying is actually true!, when Donald Trump says all mexicans are rapists, that is another thing, he is not being funny, he is lying and he is promoting hate, two very different things, and to put both things in the same category is a mistake, and I belive that is what people are doing right now, the job of a comedian is to make you laugh not to promote hate, they use stereotypes because most of them are based on true facts!
I'll tell you my opinion, and again, this is just my opinion, white people (and I am white btw, but I am Mexican so I don't carry all the baggage you do) are so scared of offending anyone, they feel uncomfortable for everything, even so that when someone else does something they consider inappropriate, I just think they are projecting their own insecurities onto them. For example, cultural appropriation, just because you use a Mexican sombrero you are now accused of "cultural appropriation", what? do you think us Mexicans care if you use a Mexican sombrero? no, its the white SJW who care. I recently read a story on the news that Kim Kardashian is being accused of cultural appropriation because she released her own brand of tequila, she basically came to Mexico, tried several types of tequila and decided to launch her own brand of tequila, apparently its quite expensive, but she is now being accused of cultural appropriation. Man this thing has to stop!, don't know if you know this but the birth control pill was invented by a Mexican scientist here in the 50s, but got commercialized till the 60s by a US scientist, is that cultural appropriation aswell?
Jordan Peterson who is a clinical psychologist and teaches at the Toronto University has said that in the future, universities will be just about STEM fields because the humanities are being so corrupted by the SJW that you can no longer say anything
Again, when did feelings became more important than facts, here in Mexico we call this generation the "Crystal Generation" because they are so sensitive that anything offends them even when you are not offending them.
I am going to tell you a story, this is absolutely true, I can point to you to the videos and the documentation, a woman professor who was teaching a law subject via remote conference right now during the pandemic to some accountant students in a university here in Mexico, said something like "you need to include Bibliographic references in your homework", the students didn't do it and argued that the term "bibliography" was something they didn't understand because they didn't study law which was the thing the teacher was teaching, arguing that "bibliography" was a law term, the teacher had to explain them that "Bibliography" is not a term used in law, its some commmon sense and something you learn in primary school, you can watch the 10 minute video of the students arguing with the teacher, the patience of this teacher is stellar. Well to be short, the students reported the teacher and she got fired, Obviously the video caused outrage in Mexico, how was it possible that these ignorant students didn't know what Bibliography meant, and overall that the teacher was fired, I don't know the end of the story, but it tells you the state of things right now.