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Yes, the kid can sponsor his/her parent working on an H1B visa in the USA for a Green card, but only if he/she is 21 or older.
Even if the kid reaches age 21 and sponsors parents, he/she has to show sufficient money/income source, pay income taxes to support parents and apply on his own and fill the affidavit of support.
The sponsorship by the Adult kids is counted under family immigration (limited yearly Green cards) for non-immediate relatives.
A parent on an H1B visa with a US-born kid is an immediate relative including any unmarried sister or brother under the age of 21 and eligible to get a US green card immediately (no yearly limits and no waiting Queue).
A green card is ALWAYS available for a US citizen, which is your kid. This can prove to be a blessing in disguise for an Indian EB2 H1B worker, who can get the sponsorship from a US-born child citizen and ditch the extremely long H1B visa’s waiting queues.
In the H1B case, it is recommended for the parent to maintain the H1B status, till the time child turns 21. H1B visa status needs an employer and continuous employment to stay and work in the USA.
When the kid turns 21, he can sponsor the parents as an immediate relative.
H1B workers will not be able to stay in the US beyond the 6-year limit, on H1B unless, they have their green card under-processing and have i140 approved.