Automated high-stake decision-making, such as medical diagnosis, requires models with high interpretability and reliability. We consider the sparse high-order interaction model as an interpretable and reliable model with a good prediction ability. However, finding statistically significant high-order interactions is challenging because of the intrinsically high dimensionality of the combinatorial effects. Another problem in data-driven modeling is the effect of ``cherry-picking" (i.e., selection bias). Our main contribution is extending the recently developed parametric programming approach for selective inference to high-order interaction models. An exhaustive search over the cherry tree (all possible interactions) can be daunting and impractical, even for small-sized problems. We introduced an efficient pruning strategy and demonstrated the computational efficiency and statistical power of the proposed method using both synthetic and real data.