Kids (for the most part) take their cues from the adults.
In my house it isn't a big deal when the mother or father or nanny come and go. The kids need to be calm and relaxed regardless of who's in charge and creating drama (tears, huge hugs goodbye, excitement around arrival, etc...) just encourages more drama.
As the nanny you have the ability to control the tone of interactions - don't indulge, don't encourage, don't secretly revel in them choosing you first. Be calm and matter of fact.
And don't get too comfortable in the "it's obvious that they prefer me" mindset. This is your first nannying position. They are the first of your charges, the family is the first of your employers, and who knows how long your nannying career will be. Children love their nannies and spend a huge amount of time with them. But a classic pitfall is when attitudes develop around who the child prefers - this is never, ever good for the nanny/employer relationship.
You and the parents need to be allies - never, ever rivalries for the children's affections. The nanny will always lose that battle, often quite uncomfortably, in the long run. I'm sure you're not running around the house talking about how the children prefer you, but it's a slippery slope and the entire basis of your question, which is why it struck me.