It is confusing to learners. I don't think there is any rule that applies to these usages. Even worse, the uses don't seem to be guided by logic.
One can be at home or at a/the nursing home.
One can be at work or at the office.
One can be at college or at the university. (AmE)
One can be at McKinley Park or at a/the park.
One can be at school or at a/the school.
If the word is used as a countable noun you need an article.
If it is used as an uncountable noun you don't use an article.
"Home" as 'the word we use to indicate where we live permanently' is uncountable. Nursing home is countable.
I am at home. I go home. I am at the nursing home.
"School" as 'the word we use to refer to the institution for educating children' is uncountable. If meaning a particular establishment it is countable.
I am at school. I go to school. I am at the school.