The school admissions code is available here. The section on social or medical need says:
Social and medical need
2.27 If admission authorities propose to give higher priority to children for social or medical reasons they must ensure that in doing so they are not failing to comply with paragraph 2.16(g) of this Code, which prohibits the use of oversubscription criteria that discriminate against or disadvantage children because of their special educational needs or disabilities.
2.28 Admission authorities must not use this criterion to give a child a lower priority in obtaining a place at the school, but it is acceptable to give higher priority to children or families where there is a social or medical need (for example, where one or both parents or the child has a disability that may make travel to a school further away more difficult).
2.29 If using this criterion, admission authorities must give a clear explanation of what supporting evidence will be required ? for example a letter from a registered health professional such as a doctor or social worker ? and how this will be assessed objectively. Admission authorities? decisions must be consistent and based on this objective evidence. The supporting evidence should set out the particular reasons why the school in question is the most suitable and the difficulties that would be caused if the child had to attend another school. Admission authorities must not give higher priority to children under this criterion if the required documents have not been produced.
(emphasis added).
So, LEAs are allowed to take account of parents' disabilities as well as children's when considering social and medical need but you would need to check what your LEA says in its admission criteria (which should be available online in their school admissions booklet).
If you have recently moved to the area and missed the application deadline, you will be treated as a late applicant but if you are accepted as having a social/medical need for a place this should place you near the top of the waiting list if you don't immediately get a place at your preferred school. (Waiting lists are - or should be, some LEAs get this wrong - held in admissions priority order).
I'm assuming this question is about admission to Reception. For older children (ie KS2 where the class size limit of 30 does not apply) you may be able to argue successfully at appeal for your child to be admitted even if the school is 'full'.