This paper presents a parallel algorithm for the following problem: given a simple graph G(V,E), color its edges with max2^{i − 1} * (Δ/2^{i − 1} + 2),2^{i − 1} * (Δ/2^{i − 1} + 3) colors, 0 i logΔ − 1 such that all edges sharing a common vertex have different colors, where Δ ( = Δ(G)) is the maximum vertex degree, and ¦V¦ = n, ¦E¦ = m. This algorithm runs in O((Δ/2^{i − 1})^3.5 log³ Δ log n + (Δ/2^{i − 1})³ log⁴ n) time using O(maxn²,n(Δ/2^{i − 1})³) processors. Particularly, it only requires O(Δ^1.75 log³ Δ log n + Δ^1.5 log⁴ n) time and O(maxn²,nΔ^1.5) processors when we use Δ + √Δ colors to color G's edges. If we use 2.5Δ colors, it only requires O(log n log Δ) time and O(m) processors, based on a CREW PRAM. Unless otherwise specified, our computational model is a COMMON CRCW PRAM in which concurrent read and concurrent write are allowed. In addition, only writing the same value into a memory cell is successful when there exists write conflict.