  We obtain matching two sided estimates of the heat kernel on a connected sum
of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key
result is the on-diagonal upper bound of the heat kernel at a central point.
Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal
behavior of the heat kernel in our case is determined by the end with the
maximal volume growth function. As examples, we give explicit heat kernel
bounds on the connected sums $R^2#R^2$ and $R^1#R^2$ where $R^1 =
R_+\timesS^1$.
