My parents are nationals of a non-EU country. I am an EU national going to live in France (and I am not French).
I am wondering whether I can invite them (i.e. ask French authorities to issue visitor visa) while I stay in France?
If your parents require a visa (that would depend on their citizenship), then they will have to apply for it themselves at the relevant French consulate. You could probably prepare the application and mail it to them so they just have to submit it to the consulate but you can't apply on their behalf from within France.
If they are going to stay with you and you are going to sponsor them, you need to get a specific form (called “attestation d'accueil”) from the administration of the town where you live. Once you have the form (signed and legalized), you have to send it to your parents who can join it to their application. The procedure is detailed in As an EU national just moved to France, how do I get a "Attestation d'accueil" for a friend's visa?
The only difference between EU citizens and others for this purpose is that third-country citizens who want to invite someone also have to prove they are themselves in France legally by showing a carte de séjour, whereas EU citizens do not. You only need to show your passport or national ID card.