You are under no obligation to share the reason for your absence. Even if you’re on parental leave, “on leave” or “out of the office” is sufficient. Unfortunately, discrimination against pregnant people and parents happens, and if you don’t want to disclose that you’re on parental leave, you don’t have to.
Let’s be real, the majority of the thousands of emails you return to after being O.O.O. will be spam and salesy marketing drivel – any legitimately important emails will probably get lost! Unless you’re Barack Obama, just send it when they’re back.

I say this as someone who used to have a chronic problem keeping up with my personal voicemails. But I got voicemail transcription set up so I can read them now, because just ignoring important phone calls has consequences. I can’t imagine trying to just duck them in a professional job where I had a phone number, and therefore an expectation that people can call me!
That’s exactly how I set up mine, except that ordinarily I’d have several options, like: “If you need help with X, call A. For help with Y, call B. For help with Z, call C. For anything else, call my manager at extension 000.