I completely agree that the shorter version, such as presented by Alphagetti is better as it is clear and concise and the parents more quickly get to the important parts of the letter and how it applies to them.
I would personally either omit the first paragraph or shift it to the last as it is not the main reason for the letter, so I find it shifts the focus of the letter.

Other suggestions include the following:
- adding that a copy of the illness policy will be posted at the entrance for anyone to review
- change this "Coughs will be dealt with seriously" to "The policy is in place to protect all our children and minimize the risk of spreading important illnesses. Although coughing often seems to be a less important symptom, it could be a sign of important illnesses, which you know xxxx and xxx are more succeptible too because of their condition. Thus, coughing will also have to be addressed in a serious manner" I fear that the previous way in which this was presented could be interpreted that other illnesses are not as important and could slip by...
- I would finish the letter saying that you are available should anyone have any questions or concerns regarding the points addressed above
- I would change this sentence "This is to protect all the children in my care."

I hope this helps. I worked as senior occupational therapist in a medico-legal environment before becoming a caregiver... which may help explain my "anal" approach to reading and writting a letter!!

It is always hard to auto-correct, especially when it is something very personal!