I believee it's the Cretaceous Period?
All of the dinosaur periods are at the start of Earth's timeline, and "the end of life's start" suggests the end of dinosaurs, meaning, when the asteroid hit. So it's probably one of the Mesozoic periods.
The T-rex first appeared in the Cretaceous Period, and the asteroid hit somewhat during middle of the Cretaceous Period.
I hope I'm right, and I hope this helps!