Virtual power plants (VPPs) promote the high-level integration of distributed energy resources (DERs) through complementation and aggregation. To avoid the privacy leakage and enhance the market efficiency, this study aims to achieve the non-iterative market participation of VPPs by accurately characterizing their external bidding functions. To this end, an improved multi-parametric linear programming (MPLP) method is developed to derive the bidding function representation of VPPs. The proposed method projects the detailed internal model of VPPs onto the point of common coupling (PCC). An algorithm for determining a precise initial parameter space (IPS) is proposed, thereby avoiding the redundant solutions that typically occur in traditional MPLP method. The IPS is then partitioned into several critical regions (CRs), within each of which the marginal cost remains constant, facilitating a piecewise linear mapping between the trading power and bidding price. Case studies on modified IEEE 33-bus and IEEE 123-bus systems demonstrate that the proposed method accurately characterizes the bidding functions of VPPs and substantially improves the efficiency of their non-iterative market participation while ensuring data privacy.