When you say "street", I assume you mean a public street... which you have to be legally licensed and tagged to drive/park on. If your tags aren't in order, you are 100% at fault being parked on the street regardless of your intentions. I've had a knock on my door before for parking an old truck in front of my house with expired tags - the cops just asked me to move it or get the tags in order.
It seems that there’s a patchwork or laws and ordinances with varying degrees of intrusiveness, but it does look that - in principle - you may end up having your car towed for having it parked with expired tags/plates on the street (and, in some places, even in your driveway).
Having said that, the degree of enforcement obviously varies. In TX, you are supposed to have a sticker on your windshield showing the expiration date. I find these aesthetically displeasing and don’t attach them.
Now, I get in a discussion about my driving with Officer McFriendly about once a year - I figure it’s just the cost of doing business, and given that I drive about 20k miles a year, an annual speeding ticket is effectively a tax on my ignoring the speed limit on the highway.
Not once did I get any questions about the registration- they can always check it on their computer anyways.
So, I guess in some places it gets quite dicey.... It may have also something to do with the neighborhood you live and the car that you drive...