Is the husband's delay in asking about his parents considered an act of disobedience (ʿuqūq), and could the wife be a cause of that, especially if she asks him to communicate with them, despite their agreement that she not communicate with them due to their stances and behaviors, which she sees as far from religion, and which include their annoyance at her not shaking hands with non-mahram men?

If the husband maintains ties with his parents by calling them as much as he can, he is not considered disobedient or a severer of ties. A short delay in calling is not considered disobedience or abandonment, as the matter depends on custom. If the wife adheres to Allah's law in her dealings with her husband's family and encourages him to maintain ties with them, there is no sin upon her, even if they dislike her.