Is it okay to parse grey?

Depends on your views regarding raids/DPSing. Grey parses are like.. the bottom 20% of the playerbase... they are, I suppose, technically bad players.

Or is it something to be ashamed of?

If you enjoy the game regardless of what anyone says/flames you with? No.

Want to actually improve/know the basics of the class(es) you play/know the mechanics of the raid? ....Yes it probably is.

She doesn't take criticism well at all

If she wants to get into harder content, to play with you... she's either got to: a) actually want to/start to improve or b) accept the flames that go her way because of how badly she plays.

is there anything i can do it get people to leave her alone?

Hiding her FFlogs profile will possibly stop flaming from the get go, from people who check logs preemptively.. but you'll probably get asked why her profile is hidden... Also doesn't stop people running ACT personally and seeing her DPS numbers for themselves in-game.

Is parse color really that important?

For people who parse as 50%+, yes. They don't want to be dragged down, or carry dead weight. I personally tend to parse purple+... and I've left groups before, after checking out the FFlogs of other people in the party, to find out that most of the group parses grey. Those groups end up being a horrific shitshow 99% of the time and I don't particularly like having to lower my own personal DPS to baby players who think AoEs are swimming pools and tanks who can't mitigate properly or scream about "no heals" when they were the ones who ignored pretty simple mechanics. I am apparently a fairly competent player, I want to play with other competent people.