Were Alī ibn Abī Ṭālib and Husayn ibn Ali alive at the same time?
Yes. Alī ibn Abī Ṭālib and Husayn ibn Ali were alive at the same time for 34 or 35 years, from AD 626 to AD 661.
Alī ibn Abī Ṭālib was 26 when Husayn ibn Ali was born.
Husayn ibn Ali was 34 or 35 when Alī ibn Abī Ṭālib died, aged 61 or 62.
Alī ibn Abī Ṭālib
4th Rashidun Caliph (r. 656–661) and first Shia Imam
Born 17 March AD 599 · died AD 661, aged 61 or 62
Husayn ibn Ali
Grandson of Muhammad and the 3rd Shia Imam
Born 9 January AD 626 · died 10 October AD 680, aged 54