Target-based LiDAR-camera extrinsic calibration is a prerequisite for multi-sensor fusion in robotics. However, in the widely adopted four-hole pipeline, \rt{calibration accuracy is limited by hole-center extraction in the LiDAR side, where sparse angular coverage and mixed-pixel returns displace the estimated centers}. This paper presents P$^2$Calib, which exploits \textit{pattern priors}, geometric constraints specified by the CAD model of the target board, to improve calibration accuracy. First, we incorporate the known hole radius as a fitting constraint to prevent center estimates from degrading under sparse angular coverage. Building on the improved hole estimates, we further enforce the rigid rectangular layout of the four holes as a global consistency constraint to correct residual errors across holes. Both priors are integrated into an \rt{interactive tool that runs the pipeline from target detection to the final extrinsic}. Experiments on simulated and real datasets show that P$^2$Calib reduces the joint registration residual by 90\% and 82\% and the held-out reprojection error by 96\% and 77\% over the baseline. \rt{We will release the code and data\codelink{} to facilitate future research.

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