Unbonded prestressed reinforced concrete (UBPRC) column has attracted the attention of many researchers in bridge seismic design because of its small residual displacement. However, the research on the mechanism and quantitative method of its seismic response under near-fault impulse ground motions is still insufficient. On this basis, a quantitative probabilistic mathematical model was established to evaluate the seismic demand of UBPRC columns with different structural parameters subjected to earthquakes with varying pulse parameters. Firstly, in order to describe the seismic response of UBPRC columns under pulse-like ground motions accurately and conveniently, a new parameter (xT) composed by fundamental period (T1) and pulse period (Tp) was proposed. It is found that Gaussian function can well describe the relationship between the newly defined parameter and seismic demand, based on a set of seismic response databases of UBPRC columns with different heights under impulse ground motions. Secondly, considering that the peak pulse velocity (Vp) has a significant impact on the seismic response of bridge structures, the influence of Vp on the parameters of Gaussian function was studied, and the functional relationship between these parameters and Vp was established. Finally, a novel probabilistic seismic demand model (PSDM) considering structure parameters and pulse parameters was established, and its accuracy is verified. The quantitative mathematical model with multi pulse-structure coupled parameters can be adopted to conduct the seismic fragility analysis of UBPRC columns. The verification and application indicate that the proposed PSDM model can quickly and accurately estimate the seismic demand of UBPRC column under different near-fault impulse ground motions.