I think what matters is the underlying approach to power and control and how you fare in that negotiation. In most employment situations, the worker taking unscheduled time off (to care for sick kids) will eventually cause problems, whether the worker is part-time or full-time. The employer of a part-time person presumably needs that person present during their scheduled hours, even if the total weekly hours are not full time. Just because your husband earns more money does not mean that you taking off unscheduled time for sick kids won't impact your employment.
So what he is really saying is that: it is acceptable if you lose your job because of taking so many sick days to look after the kids.
So is it? It may be acceptable to him, but is it acceptable to you?
It sounds like you are already in a situation where your time and effort are not valued because your on-the-books income is lower than his. Is your "in kind" contribution of childcare valued? Or is it just treated as "Oh well you're home you'd have to look after them anyway." Is the care of your children treated as a nice hobby that is just yours, rather than a moral obligation that is half the responsibility of each of you? The fact that your salary "pays for childcare" and "bits" suggests the former. Try thinking of it differently. How much would it cost to buy your services on the open market? The best comparator for your services isn't a babysitter or a childminder. It's a nanny. And actually a mum does more than a nanny because a nanny probably won't deep clean the toilet while also watching a child. Your work is valuable and would cost a lot to replace!
If you lose your job due to taking too many sick days, you will become completely financially dependent on him. How do you think that would go? Would he be respectful? Would the earned money be just as much yours as hiis? Remember, if you quit, it would be SPECIFICALLY because he wants to keep his earning power and does not want to compromise that for sick days. So even if you never earn another penny, it would be YOUR childcare that allows him to maintain and increase his earning power, and it would be HIS refusal to participate in his half of the childcare that prevented you earning your own income. Would he recognise that? Or would he just come to see you as a bon-bon-eating deadweight? Will he be willing to equally share any disposable income on your needs and wants as much as his own? Or will you be watching him go on nights out with his buddies while you can't afford to get a haircut or a new coat? Will he even recognise that you have needs? Or will he dismiss your claims that you need him to babysit so you can see a doctor as "Oh you're not that sick, you'll be fine."
Remember that this can change slowly over time. The man who initially seems to divide things fairly may come to see you as a deadweight when you are both older and he has increased his income to a high and stable level, whereas if you've been out of work for a decade and can only get a minimum wage job, he may think of you as only being "worth" minimum wage in the relationship. Even though that change in your earning potential was a direct result of, again, leaving the job market not out of laziness but because he wanted to grow HIS career and didn't care about yours.
Only you can make the best estimate as to the answers to those questions. But I think him wanting you to quit is not a good sign. Just my opinion.