Is there any option to adjust the status of an adult child of a U.S. citizen and LPR to immigrant under his parent’s sponsorship?

An adult child does not qualify as a dependent. It is not clear under what process you are seeking to include the child so if you clarify , perhaps the answer would be more specific to your question.

If your son is an adult unmarried over 21 years, and as a US citizen, you can file a Form I-130 petition for him. Note that the visa will take about 7 years to become current under the F-1 preference category. [Adult unmarried son/daughter of a US citizen] Note that in that category, a visa will not be current for at least 7 years. Filing a petition for him does not grant him a legal right to be in the country.

You can sponsor your son for family-based immigration by filing Form I-130 immigrant petition. However, filing of a petition does not give him the right to stay in the U.S. until he gets a green card. And it is extremely important that he should remain in a legal status, because, if he stays in the U.S. illegally, he will not be able to receive a green card even through your petition. To stay here while his immigration process goes through, he would need to a) extend his B2 status, or b) change his status to some other non-immigrant visa - F1 student visa, or H1B work visa, or any other non-immigrant visa he can qualify for. You also need to understand is that your son will not be eligible to adjust status (receive a green card) any time soon. Every year, a limited number of immigrant visas is issued - 23400 to unmarried adult sons and daughters of U.S. citizens (F1 category); - 26266 to unmarried adult sons and daughters of LPRs (F2B category); and - 23400 to married sons and daughters of U.S. citizens (F3 category). Many more petitions get filed every year, and there is waiting time - right now in F1 category - 7 years (12.5 years for natives of the Philippines; 20.5 years - for natives of Mexico); in F2B category - 6.5 years (11 years - for Filipinos, 21 years - for Mexicans); in F3 category - 11 years (21 years for Filipinos and Mexicans.) This means that the earliest time an immigrant visa might open for your son is about 7 years away. Extensions of B2 status even for 6 months are quite difficult to obtain; and there is no possibility to extend your son's B2 visa for 7 years. Besides, even if it were possible, your son would be unable to work all this time because accepting employment violates B2 status and makes him deport-able. So, your son has 2 options: a) he can talk to an immigration attorney to figure out whether there is a non-immigrant visa he could get to stay in the U.S. until his turn for an immigrant visa comes; or b) he can leave the U.S. when his admission expires and wait for an invitation for an interview at the U.S. embassy in his native country.

If your son is still under the age of 21, he is considered an immediate relative and may apply for adjustment of status concurrently with the I-130 petition filed on his behalf by his US citizen parent.