If anything, having fewer/no children is a mark of status, both globally (higher income countries tend to have lower birth rates) and in the US (vaguely implies both parents are working and have access to contraceptives, etc).
To be clear, I don't think that there's much of a status signal either way.
21 percent of families earning > $200K had at least 3 kids in 2011, an increase from 15% in 2000. Every other demographic saw their numbers go down.
https://www.nytimes.com/2014/04/10/fashion/The-Growing-Three...
Anecdotally I see that too. Almost all of my children's classmates come from 2 child families with the occasional single; almost all of my wife's surgeon friends have 0 or >2 children.
I'm applying the following standard: if someone told me that their friend had 3 kids, would I infer something about their social status? At least for me, not really? Like you said, the poor and rich both have >3 children. ymmv of course