Regarding 'who pays?', there (as ever) is a process in place. Some schools employ a cover supervisor. But all schools should have a budget allocated for cover. Managing the budget is the head teacher's job - any head without a budget for cover would be seriously financially mismanaging the situation. The third option (for absences due to favours, etc) is internal cover where you ask colleagues a favour for something that you aren't entitled to time off for. Allowing this is at the discretion of the head.
Teachers get sick, need to meet with social services, need to do job interviews, break their legs, get stuck in a snow drift, have funerals of loved ones to attend and so on and so on. And even have important hospital appointments. These absences are inevitable and have to be covered by the budget.
Of course, budgets are being slashed for schools at a frightening rate, and head teachers are trying to economise everywhere. Cover budgets is an obvious one to address. I understand why heads are reminding teachers that absences cost money to the school.
However: teachers also take time off with stress and depression, fatigue-related illness and so on. Or when they are interviewing frantically for jobs elsewhere, or throwing sickies to be able to actually manage their lives. These will end up being paid for from an emergency cover budget as well, but any good leadership should take all steps available to them to minimise these absences through developing a healthy environment to work in, just like any other business. And schools ARE a business, and an employer. Our profits are results and well-nurtured students, but it is still a business nonetheless.
Like a majority of employees who work in vocational jobs, most teachers go way beyond what their job requires them (and pays them) to do. For this, our county is lucky. However, encouraging this culture where teachers do not believe they are entitled to time off for things they are legally entitled to have time off for is a short-sighted nonsense which creates resentful teachers and wary parents. No-one wins.
I am very thankful that my GP squeezed my DD in for an emergency appointment between surgeries because I was worried. I know this was his lunch-time, or admin time, and that he did not have to do it. That was a favour and is over and above what's expected. It is much appreciated. Ditto the midwife who stayed late on her shift to deliver DD, the home help who popped out on her day off to see my DG, etc etc. I am glad to say that I could go on and on. These people are doing vocational jobs. And yet! I would expect a locum GP, my midwife to be changed, etc if their child was in hospital or their spouse was ill or they were at their DM's funeral. It happens and we have systems in place to provide for it. Use those systems or we lose them, and then it really is an impossible job where the most vulnerable suffer.
Anyway - enough of my essay! Your DH is experienced and established enough to man up and ask his HT for a day off. Again, I suspect this is not really the issue.
Hope the allergy challenge goes well.