You should call and confirm since PA is relatively new field it is sometimes not posted in advertisement but the words like "related field" may turn in your favor as both are similar fields to some extent .
It won't hurt to confirm.

You can apply as Asst. Director ( A & E ) as requirement is Second class or Grade C 16 years of education in Business administration HR/ Management or PUBLIC ADMINISTERATION with 2 years experience.
As for Asst. Director (Finance and accounts) you should confirm