My first thought is maybe it is more important.
I mean dh earns about 3x what I do (both full time), has far more responsibility than I do, far more opportunity to advance etc. So if for some reason one of us had to give up our job for some reason, there would be no question who did.
Neither of us would have any doubt. I'm not sure we'd even really discuss it.
If I said to dh that "someone said you think your job is more important than mine" then he'd probably say "yes it is." Tbf I'd probably have said to the person "yes it is too!"
However his job is more flexible than mine and he works from home a lot of the time, so 95% of the time now when one of us has to flexi, it will be him. However if he has a unmovable meeting or is meeting a client etc. I would be the one that rearranges my work round it.
But if you're working from the same place then presumably you have the same flexibility etc.
So what I'm wondering if it's things like childcare. If (when the dc were younger - and tbf I was in a more flexible job too due to being part time, and he wasn't wfh) I would always be the one who came out of work to collect/appointments/illness etc. And we'd both have known that. If the school had phoned dh, then he's have phoned me to collect. If there was an appointment, I'd have the time off. If the gasman was coming, then I'd be the one nipping home to let them in and going back to work afterwards.
So if she's noticed that you do all the child appointments; are the one who stays at home to let the workman in, and maybe he's said a few things that have implied to her that he would expect you to do it with no thought that he might, then that might have inspired that comment. And there's nothing wrong if that's the way you did it, but it may have given that mindset to your dh without him thinking about it.
I've got a funny story about how the mindset can work. As I said, before dh wfh I would nip back for workmen etc. I was part time, so could make the time up and I worked 10 minutes away. I'd get them to call me when they were on the way, and drive back. Worked well.
Once dh was working from home after lockdown, I'd booked the gasman for a service. The conversation went like this:
Me: The gasman is coming to do the service on Wednesday about 11am,
Dh: Why are you telling me?
Me: Because you'll need to let them in.
Dh: What if I'm in a meeting?
Me: Have you got one booked?
Dh: No
Me: Then if someone asks for one, say you have to let the gasman in at that time.
Dh: I'm not sure I'll be able.
Me: You say "sorry, just got to let the gas man in" to your meeting, walk to the door, say "hello, there's the boiler, let me know if you need anything, sorry I'm in a meeting, so I need to get back" and go back to the meeting. 2 minutes.
Dh: Won't you be coming home to deal with him?
Me: You think it's better that I drive 10 minutes from work, wait around while he does the work and then go back, and have to make about 90 minutes of work up at the end of the day when you can take 2 minutes out of your day? On the basis you might have a meeting?
Dh: Ah!
Tbf to him he has done it since without any worries (and even comes out of meetings as long as they're not with clients). There are occasions when I point out to him how much better it would be if he dealt with the tradesmen and arranged them for times that suit him, but until he does, then he'll have to put up with my chosen times. And tbf he's not very good at that sort of thing because he knows nothing about it, so it's normally better I say what I want and he just lets them in.
Dh has done a lot of the cooking/shopping/childcare etc over the years (now does most of the cooking and shopping, no childcare needed) even when I was sahm so it's not that he's unwilling, just that it didn't occur to him that he needed to do it because I always had, and he hadn't really thought how it effected my day as opposed to his.