So happy for you!
Entering into an established polyship, I feel that respect and space are two of the most important things you can give to your brand new metamour. She probably feels as if not more threatened than you, and is wondering how their relationship is going to change as a result.
And once you allow your new metamour to come sniff you out at her own pace, and she gives you the assuring lick on the nose (what? how do you mean I spend too much time around rabbits of late?), the two of you can start the ooshy-gooshy metamour crush phase.
NRE turns perfectly lovely people into monsters. It also tends to drive people to definitely more monogamous interaction patterns. This can be really hard to the established (I prefer this term to 'primary', which I connect more with household economics than anything else) partner. Hence the respect and space.
The one thing your parents will be worried about, if this relationship progresses and you out yourself to them, is if you are being used. Responsible non-monogamy as a concept goes pretty much against everything we as a culture are taught to believe is right and good.
Me: bi female in my twenties