Rethinking Newton's $\textit{Principia}$

It is widely accepted that the notion of an inertial frame is central to Newtonian mechanics and that the correct space-time structure underlying $\text{Newton's}$ methods in $\textit{Principia}$ is neo-Newtonian or Galilean space-time. I argue to the contrary that inertial frames are not needed in $\text{Newton's}$ theory of motion, and that the right space-time structure for $\text{Newton's}$ $\textit{Principia}$ requires the notion of parallelism of spatial directions at different times and nothing more.