Inference on a scalar parameter in the presence of high-dimensional nuisance parameters is challenging, especially in a scenario where the dimension of the nuisance component increases with the sample size. This difficulty intensifies in models with crossed fixed effects and sparse discrete data, where standard likelihood-based methods often prove unreliable due to limited information. While parametric bootstrap methods present a viable alternative to analytical corrections, their application to sparse discrete data and crossed fixed effects remains underexplored, with theoretical properties yet to be fully established. This study demonstrates that constrained bootstrap methods leveraging penalized estimates outperform traditional bootstrap approaches, particularly in logistic regression models under severe sparsity. These findings highlight the potential of penalization strategies to advance inference in models with high-dimensional nuisance parameter structures.