If you remove LLL part then is remaining part of
a. Lenstra algorithm
b. Barvinok algorithm
in $O(f(n)(\log(mL))^c)$ time on $O(g(n)(mL)^c)$ processors with fixed $c>0$ in fixed $n$ dimension, $m$ constraints and $L$ bit length input for some functions $f,g:\Bbb R\rightarrow\Bbb R$?
Note we know linear programming and convex programming are not in $NC$ in general dimension.