It is not permissible to fast the 13th Day of Thul-Hijjah
Question :
When the fast of the Days of Whiteness (Ayyamul- Beedh, i.e., the 13th, 14th and 15th of each month) coincide with the days of At-Tashreeq (the 11th, the 12th, and the 13th of Thul-Hijjah) is it permissible to fast or not?
Answer:
It is not permissible to fast the 13th of Thul-Hijjah, whether it is a voluntary fast or an obligatory fast. This is because they are days of eating, drinking and remembering Allah, the Mighty and Majestic. Indeed the Prophet forbade fasting them and he did not permit that for anyone other than whoever could not make the sacrifice of Hajjut-Tamattu'. In this case, such a person is allowed to fast the 3 days of At-Tashreeq in place of the sacrifice and he fasts the 7 remaining days (i.e., 10 all together) with his people (i.e., upon returning to his homeland). This is due to what has been confirmed in Sahih Al-Bukhari from 'A'ishah and Ibn 'Umar that they said: "It was not permitted to fast the days of At-Tashreeq except for whoever could not make the sacrifice." In reference to fasting the 14th and 15th (of Thul- Hijjah), there is no harm in that, because they are not from the days of At-Tashreeq. And success is from Allâh.
Source:
Ash-Shaykh Ibn Baz
Fatawa Islamiyah, Vol. 3 Pages 341-342