yabu to regard time spent caring for children when they are ill is a 'waste' of annual leave.
children do get ill from time to time. when they are ill they mostly need parental loving care.
a full time job requires your time for generally less than half the day merely 230ish days a year. your annual leave allowance is sufficient to cover a couple of weeks away plus ad hoc days as needed to cope with the ebb and flow of normal family life - these will not all be nice days of holidaying but will include days of waiting in for the plumber, attending funerals of relatives too distant to qualify for compassionate leave, caring for sick children and dozens of other calls on your time that can't be scheduled for the weekend.
a decent employer won't punish an employee for having to take time off at short notice when a child is sick (as pp said make sure they know you are only taking your fair share split with dh). a decent employee will not take the piss by trying to get around the inescapable facts.
pretending to be sick yourself is lying, effectively stealing and should be treated as Gross Misconduct.
working from home may sometimes work for some jobs but doing this too often isn't going to be sustainable and there will be a lot of things you can't do when not in the office and it's not unreasonable for an employer to be unhappy with this as a main strategy.
use your annual leave for this.
if you then end up with insufficient annual leave in-hand for family holidays come summer, you can book that week as Parental Leave (which is unpaid but needs to be agreed in advance not just taken ad hoc).
if you are unlucky enough to have a child whose health is so precarious the days off work for caring begin to overwhelm the available leave you and dh may need to reconsider whether both working ft is sustainable.