He was afraid that his parents would be angry and disappointed.
Traditional grammar calls that clauses 'noun clauses', but modern grammar claims that term is a misnomer because they don't behave like nouns. For example, they don't normally function as extraposed subject or as complement to a noun or adjective.
Nowadays, scholarly grammar calls them content clauses because they serve to give or expand the meaning, i.e. the content, of the word they complement.
Content clause is the default kind of finite subordinate clause (the others are 'relative' and 'comparative' clauses).
In your example, the clause is functioning as complement to the adjective "afraid".