I try to be as reasonable with my sick policy as I can as I know that it's hard for parents to take time off work but they keep sending in these kids who look like death just ran them over.
That is the problem.

Sick policy is black and white. Because you worried about lost income due to personal situation, you allowed area of grey "trying to be reasonable".

Parent is getting message that you ignoring own policy so why should they follow it. Take step back, start enforcing the policy and then it clear message in document and in person that is policy not guideline.

Where you mention being single parent, imply you maybe not turn away sick kid to keep income. If you don't charge when kid sick, change that protect your income and might help you be firmer. Just idea.