Manusmriti is 2 century BC whereas Mongol invasion is middle ages and had very little impact in India because for various reasons the Mongols did not really get inside India although there were a few battles where some Mongol generals tested the borders.
Bhakti movement, a reformist movement of Hinduism mentions caste, thats 8th century AD, well before Mongol invasion. There is mention of caste in Budhdhist, Jain and Sikh literature as well. A significant component of many of these religion were a rejection of the prevailing caste system.
From the sound of it, the parent's claim has a whiff of what is derogatorily referred to as propaganda of WhatsApp University -- a well oiled and funded propaganda and fake news machinery of the political party in power. Please do not fall for it.
1) Mongols never invaded India. Even though they reached river Sindh while hunting a Muslim ruler. They had deference towards India - the land Buddhism arose from. Some historians say lack of grassland etc but they never came here. Even your Wiki link says the same except for the disingenuous headline. Kashmir was invaded my Muslims from Mongol captured regions not Mingols themselves. Mongols were not Muslims.
2) Show me the Sanskrit text of manusmriti talking about anything but the varna system.
I have studied this in original Sanskrit and dont need cut/paste reference to bolster common but false propaganda from Wiki or WhatsApp university. Satmaye Jayate - truth is always victorious.
> Mongols never invaded India. Even though they reached river Sindh
I agree with you and havent claimed otherwise (quoting myself: Mongols did not really get inside India although there were a few battles where some Mongol generals tested the borders.). Was responding to parent's claim that caste system was an artifact Mongol invasion. Not sure what relevance of them not being Muslims or the scarcity/search for grassland (both true statements) have anything to the topic though.
> Show me the Sanskrit text of manusmriti talking about anything but the varna system.
The word caste of course cannot occur in a sanskrit text. By caste one means the varna system of designating people into Brahmins, Kshatriya ... and others
The varna is about individual callings or occupations, not caste as understood today in India. It also has its correlates elsewhere, e.g. in the Confucian "four occupations of the people" and the tripartite division in the West. Would you say that Europe had a caste system historically because of the definition of First, Second and Third Estates?
Even the Govardhan Mutt Shankararcharya - one of the custodians of Advaita, argues clearly that the varna and jati are integral to Hinduism and he supports the caste system. You can find his videos on Youtube.