This is incorrect. The children do not get a benefit check if the parent is receiving SSI, only if the parent is receiving SSDI or retirement SS.
SSI is welfare for those who are disabled but who didn't work long enough to get enough credits to qualify for SSDI. SSI is dependent upon household income and assets, and there is no extra check for the children.
SSDI is for those who are disabled and who worked long enough to get enough credits to qualify for SS. Household income and assets are not counted, and there is a benefit check for the children.