This incident proves that the prohibition was abrogated since the Hadith in which the Prophet (S.A.W.) prohibited the usage of the kunya was prior to the one in which he gave permission for its usage, hence the permission coming after the prohibition abrogates it.
Moulana Muhammad Yusuf Binnawri (d.1397 A.H.), a commentator, on the Jami’, said that Imam Tirmidhi was born in Bugh in the year 209 A.H. He belonged to the Banu Sulaym tribe, hence he was called Sulami. His native town Bugh is a few miles from Tirmidhi and is considered a suburb of Tirmidh. Therefore he was known as Tirmidhi and Bughi, both places being in Khurasan, which is presently a province in North Eastern Iran, and it is with the nisba (linking him to his place of origin) Tirmidhi, that he is well known. The aforementioned biographer of Imam Tirmidhi does not mention any details of his parents. Imam Tirmidhi is reported to have said that his grandfather belonged to Marw but he subsequently moved to Tirmidh. Likewise, no mention is made of Tirmidhi’s early life or occupation.