One of the 6 questions of Socrates, the other 5 are in their own topics.
I think you posed a question and used available resources ( dictionary ) to answer said question. Now thanks to your post we all know the definition of piety, thanks!
If there are not gods, is piety a valid concept?
Definition #3 involves dutiful respect for parents or homeland. If the homeland is abusive, corrupt, and has unjust laws and society, it piety towad that homeland a virtue or a vice? If parents are abusive, selfish, do not nurture their children or prepare them for a successful life, is piety toward those parents a virtue or vice?
The word might not be valid outside of religion, but the concept may apply.
Could it maybe be about being true to your values or ideals, e.g. religion for some (hopefully fewer), humanism, objectivism or being ethical?
Piety generally means respect for and observance of the usages and practices of religion. Religion is involved with that which is sacred and a whole group of words including: sacred, holy, reverend, et cetera are little more than code words for DO NOT DISTURB and particularly DO NOT QUESTION.
Someone who is pious either buys into that noise or wants others to buy into it, an obvious non-starter with us. Piety is bullshit, Q.E.D.
I do think that familial piety, respect for one's elders and parents and ancestors, is a humanist virtue. Older people tend to have more wisdom and we live in a society that devalues the elderly oftentimes, youth don't listen to or care for or respect their elders. This does not mean that we must share their views, values or beliefs, but the sacrifices that my parents made for me make them, in my eyes, worthy of a VAST reservoir of admiration and respect that is well-deserved.
Hiram, I both agree and disagree with your comment. Some older people have amassed wisdom; others have spouted idiocy blindly for decades and will continue until death ends their foolishness. We should listen to them only if they have something of value to say. My mother was a wonderful woman, and I cared for her emotionally and materially until she died. But that was only in part because she was my mother; she had earned my care for her by her care for me. If she had abused me throughout life, I might have seen that she was fed and sheltered, but I wouln't have "cared" for her. (And my belief system is vastly different from hers.) So I'd say admiration must be, as you say, deserved.
Piety is an insurance policy before insurance companies existed. It doesn't necessarily have anything to do with religion at all.