What is the way to deal with parents who are far from religious matters, refuse advice, and engage in actions that violate Islamic law, especially with the fear of that influencing younger siblings, and to avoid participating in any sin?

The situation your parents are experiencing—distancing themselves from religion, abandoning prayer and hijab (definition), engaging in immodesty, disregarding halal (definition) and haram (definition), and your father's doubt about the Afterlife—is lamentable. Doubt in the Afterlife is disbelief (kufr), and abandoning prayer while denying its obligation is outright disbelief, as is abandoning it entirely according to one view. Whatever the case, they are your parents, and you must honor them and accompany them in kindness, continuing to advise them and pray for them. Do not help your mother in purchasing forbidden things; rather, gently persuade her against it.