We had a fun time at the park.
Yes, "fun" is an adjective. We can call it "informal", just as the dictionary does. However, "fun" used as an adjective is so widespread that it is impractical to say that it is not, and it is impractical to tell English language learners that a sentence such as "we had a fun time at the park" is wrong. It is correct.
The use "fun" as an adjective is well into the millions, as we can see in these quick sample searches from Google.
fun: Definition, Synonyms from Answers.com
"a fun time" - Google Search
They're fun to be around. - This is correct, and it's just the sort of thing that people say every day.
"fun to be around" - Google Search
fun: Definition, Synonyms from Answers.com
USAGE NOTE The use of
fun as an attributive adjective, as in
a fun time, a fun place, probably originated in a playful reanalysis of the use of the word in sentences such as
It is fun to ski, where
fun has the syntactic function of adjectives such as
amusing or
enjoyable. The usage became popular in the 1950s and 1960s, though there is some evidence to suggest that it has 19th-century antecedents, but it can still raise eyebrows among traditionalists.
The day may come when this usage is entirely unremarkable, but writers may want to avoid it in more formal contexts.
For me and many others, that day has arrived.