Would you invite your personal slave to YOUR birthday party? I didn’t think so.

well, he IS a baby dragon, and he probably were not really able to go along because he had to look after the library, and they would probably party late into the evening, when he would fall asleep and miss out on the fun anyhow.
 
Or maybe Spike did get an invitation, but he declined for some reason.
 
He was up all night preparing her birthday party the previous day and couldn’t make the trip.
 
Pick one. We don’t see everything, just the things that are important to the plot, that’s it. Who’s to say that in the montage they told Rarity that Spike couldn’t come? It was a Rarity centered episode, so we’ll know only what she would know, that’s it.

How do you know he wanted to go?
 
Besides, there still was a party at the library, so Spike probably stayed to party it up there. It’s the same reason none of Twilights family was there.