You can apply for naturalisation if you are over 65, and have been living in France over 25 years.
And the residence permit is a “vie privé et familiale” one. It works if children/grandchildren are minors and you have been contributing to their upkeep.
JaneJones already answered you for the most part, if you’d like to dig deeper => Les parents d’enfants français / L'immigration familiale / Immigration - Direction générale des étrangers en France - Ministère de l'Intérieur
Yes, good point. When in doubt, consult the official French govt site. Unfortunately, these days we have to verify the accuracy of what’s published. I try to do that when possible. Go to the source. Worse in U.S.A.!!!
I do have a subscription and I do get value from it, but I wouldn’t trust it. Also, too often Colonel (and Mrs.) Blimp types appear on the letters pages. One could argue that some of its regular contributors are in that category too 
Harrumph!
You wouldn’t be thinking of S. Heffer now would you?
And Frenssh she spoke, after the scole of Stratford atte Bowe, For Frenssh of Parys was to hire unknowe.
Geoffrey Chaucer (and I hope I have quoted him correctly)
I haven’t read Geoff’s stuff since I was at school, but I think you have! 
Although it also needs checking, I find that TheLocal is a lot more reliable and also more comprehensive. It’s a different kind of newspaper and a lot less doom-orientated than Connexion. TheLocal also has an excellent political journalist and I find his analyses useful in understanding the shenanigans in the French political system.
I subscribe to it and used to subscribe to Connexion so I get their daily emails with headlines in and the contrast between the two is often hilarious…
We’re also fond of the Local.
Is that Ben McPartland? I used to be in a pub quiz team with his dad!
He’s certainly there but I was thinking of John Lichfield - I’m naturally not at all prejudiced in his favour because he lives in Normandie 
Thanks Angela, I’ll try it out.
Worth a go, I think, but it does take a bit of getting used to. I tend to look at whatever appears in their daily newsletter as searching the site is, to my mind, less than intuitive.
They do a podcast every Saturday (but not I think during holidays) in which Ben McPartland, Emma Pearson and John Lichfield discuss the issues of the week. I quite enjoy those!
I see the podcasts, I might start with them.