The simple Homer algorithm solves the problem of evaluating a polynomial of degree d with n indeterminates; in this paper it is shown that its implementation on a parallel computer with $O(d)$ processors can achieve a complexity of $2\lceil {\log _2 (d + 1)} \rceil \cdot (\lceil {\log _2 n} \rceil + 1)$. If, in addition, the evaluation of all partial derivatives is also sought, then the full Homer algorithm solves this problem on a parallel computer with $O(d^{n+1})$ processors, achieving a parallel complexity of $2\lceil {\log _2 (d + 1)} \rceil \cdot (\lceil {\log _2 (d + 1)} \rceil + \lceil {\log _2 n} \rceil + 1)$.