Your time complexity of log n appears from nowhere. It is entirely possible that the complexity of calculating the original function F was worse than polynomial

Your time complexity of log n appears from nowhere. It is entirely possible that the complexity of calculating the original function F was worse than polynomial in the number of variables, in which case a polynomial/linear increase in the number of machines could not reduce the complexity by more than a polynomial/linear factor, thus leaving the computational complexity worse than polynomial (e.g. exponential/super-exponential). I've made a comment further down that posits an idea similar to yours, though with a bit more precision.