We're out of the wood with this one now our kids are older, but it has been an issue in the past. Both DH and I have been out of work at times since our kids were born, him especially as he tends to work contracts.
My longest stretch was after DS was born and I'd been made redundant at end of mat leave (along with other team members, nothing to do with me having a baby) but that wasn't so bad as I couldn't afford childcare for 2 kids on my salary anyway. Also it was psychologically quite easy because if people asked what I was doing I could say 'I'm at home with baby now, but I'm looking for a job' and no one would think anything of it because I'm a woman. But they might judge a man more for it - although maybe not now with a higher profile for paternity leave and men picking up more.
DH would feel much more self-conscious about being present at the school gates (this was pre-COVID and I think it's probably less of an issue now where you might just be WFH, not out of work). He would also, for example, insist he couldn't ask for a different interview slot due to childcare responsibilities, something I did a number of times. I found his view both annoying and interesting - as a woman, you expect people to understand that (TBH, you don't even necessarily mention the reason why, you just say 'Sorry, I can't do that time, can you do....?) but would a man be seen as unreliable if he said he had to pick up kids or do men feel asking for a different slot just sounds flaky? And has that changed since we had that discussion, probably about 8+ years ago now?
Are jobhunting mums and dads are still held to different standards about what their priorities should be and how 'OK' it is for them to not be working?