Likelihood of becoming a public charge is a ground of inadmissibility to the US. Reliance on some benefits would definitely implicate this section, where others may not. Please consult an immigration attorney for more specific information.
This could have a a negative impact if you apply for future immigration benefits. Consult an immigration attorney.
Alexus P. Sham email@example.com (917) 498-9009. The above information is only general in nature and does not constitute legal advice. It does not create an attorney-client relationship.
The parents should not state anywhere or to anyone at the benefits office that they are U.S. Citizens, or even permanent residents (green card holders). Falsely claiming you are a U.S. Citizen to get any type of government benefit is a serious issue when applying for legal status.
It is not likely that the folks in the benefits office will ask, however, this is a population they deal with all the time, especially in California.
As for the welfare issue, when a person applies for an immigration benefit, they will have to disclose any type of aid they have received. However, they also need to show that they hav e the ability to support themselves (through an affidavit of support from a USC/LPR) at that time. It is a totality of the cirmstances analysis. This might be helpful: http://www.uscis.gov/portal/site/uscis/menuitem.5af9bb95919f35e66f614176543f6d1a/?vgnextoid=775d23cbea6bf210VgnVCM100000082ca60aRCRD&vgnextchannel=8a2f6d26d17df110VgnVCM1000004718190aRCRD
This is a complex issue and you should speak with an attorney before taking any action that may affect your future rights.