It might be illegal for you to possess such data after you were asked to destroy it, yes, at least in the EU.
The GDPR talks about "data controllers", and citizens have the right to demand such controller remove their personal data. A "data controller" in this context means you knowingly possess the data and are in the position to make decisions about it. You're not a data controller, tho, if e.g. you run some service that allows users to upload data, without your involvement and direction and also do not decide how to use such data. E.g. amazon would not be liable if people put a data dumb in their cloud (unless amazon used the data themselves, instead of just storing/hosting the data at the behest of their customers).
Even before the GDPR there have been related laws and court cases, like the case that culminated in the "right to be forgotten" based on a decision by the European Court of Justice, which may well come into play here. I also remember a case in Germany, where a women allowed her partner to take intimate pictures of her, then after the relationship ended had a court order him to destroy the material (not a revenge porn case, there was no allegation he ever shared any of those pictures), meaning it's not always about what's public.
I don't know how California's mini-GDPR compares.
Then of course there is still the avenue of copyright law if the stuff is put online. Just because a parler user gave parler the permission to distribute a certain piece of content doesn't mean that everybody else has the same permission. I'm pretty sure Parler didn't make people assign them the copyright (which isn't even possible in some jurisdictions), therefore the people who posted on parler still retain the rights to their content. They can therefore use the DMCA or other similar jurisdictions around the world to demand takedowns.