To address the consensus control challenge in model-unknown linear parabolic distributed parameter multi-agent systems, this paper proposes a novel data-driven iterative learning control (DDILC) method. A forward difference scheme is adopted to discretize the spatiotemporal dynamics, thereby establishing a data-driven input-output model that eliminates reliance on precise mathematical formulations. Neural network approximates the unknown linear operators, and an error-driven iterative learning law updates the network weights online to compensate for unmodeled dynamics and uncertainties. Rigorous Lyapunov-based analysis verifies the boundedness of the weight errors and the exponential convergence of the consensus tracking errors. Numerical simulations with one virtual leader and four followers confirm that all agents achieve precise consensus tracking within 20 iterations, with errors converging to within 0.01. This method overcomes the limitations of traditional model-based control, offering an efficient solution for consensus control in engineering applications such as UAV swarms and intelligent industrial collaboration.