In the diagram i have posted, i don't see much room for changing the algorithm to execute in parallel, except maybe running the inversion logic in parallel with the last multiplication, but that gives only ~10% efficiency boost due to inversion being much slower than multiplication.
For the Y calculation, since you are not doing any addition/subtraction, you can leave out the modulus until the very end.
You can also try computing multiple points in parallel.
And I'm not sure if factoring will work, but just as how in regular numbers we can compute a factor tree, for numbers representing point coordinates, we can have them in terms of G, G*2, G*3, G*5 and so on. I'm not exactly sure how this would be implemented, but if computers were able to find
a prime number much bigger than 2^256, then it should be reasonably possible to enumerate all of the prime numbers from 1 to 2^256 and calculate the products with G, and make some kind of factor tree out of that.
The idea is you'd no longer have to do the algorithm for P, but you could do it for all of its factors instead, and multiply all of the results together at the end.