Vezenkov is the son of Yanka Vezenkova (née Gerginova) and Sachko "Sasho" Vezenkov, who has served as both head coach and director of sports of Lukoil Academic. Sasho is also a former international basketball player, who played at EuroBasket 1985, EuroBasket 1989, and EuroBasket 1991. Sasho was also the captain of the senior men's Bulgarian national basketball team. Sasho moved to Cyprus, and he played and coached basketball there in the Cypriot Basketball League for many years.