Correct spelling, explanation: in the written form, 13th reads thirteenth as it sounds the most natural. Th can be considered the most common of all suffixes among ordinal numbers, as it comes after most regular numbers (numbers ending in 4-9, and numbers 10-19). When we add an ordinal number suffix, we use the one which is used at the end – in this case, it’s -th.

Definition of 13th:
adjective
1. ordinal number of 13.
The number on my ticket said I’m 13th in the line.
The 13th international Calligraphy Exhibition kicks off in Paris this week.

Incorrect spelling, explanation: while 3rd would absolutely be a correct form (third), it is important to remember, that 13 is read as thirteen, so thirteenrd would not make much sense and would be extremely hard to pronounce, hence 13rd it’s incorrect.

Incorrect spelling

Incorrect spelling