The PDF links in that link above don’t seem to be working. The below is from Cal, which will have the same policy as other UCs- there is a list of eligible non immigrant visa categories in this link.
I’m assuming your family is most likely H1B/H4, which is on that list, but there are many on there (basically any visa that allows living and working in the US) so check if it’s something else. Do note the condition “If you hold one of the following visas, to maintain eligibility for residency for tuition purposes, you must provide documentation of your current eligible immigration status every semester of attendance.”