If we paid police officers and fire fighters with tax deductions, rather than wages salaries, it would be the same, yes. But why would we pay them in such an obscure fashion? Indeed, why do we choose to pay parents via the tax code, rather than just cutting them a check?
Because someone would immediately try to unionize parenting, negotiate 'contracts' with the government that are untenable in the long run, and in very short order collapse our economy.
If you accept the comparison, there is no way that parents could be paid anything like what the work is worth. So instead, a little tax break? Better than nothing.