I think you can solve this in O(1) time per query given O(n) preprocessing to find the locations of all 2 and 3 character palindromes. (Any even plaindrome will have a 2 character plaindrome at the centre, while any odd will have a 3 character one so it suffices to check 2 and 3.)
For example,
Given your string baraabarbabartaarabcde, first compute an array indicating the locations of the 2 character palindromes:
baraabarbabartaarabcde
000100000000001000000-
Then compute the cumulative sum of this array:
baraabarbabartaarabcde
000100000000001000000-
000111111111112222222-
By doing a subtraction you can immediately work out whether there are any 2 character palindromes in a query range.
Similarly for three character plaindromes:
baraabarbabartaarabcde String
01001000100000010000-- Indicator
01112222333333344444-- Cumulative