This is clearly a problem on Snapchat's end, not Spotify's.
It happens with Spotify, Podcasts, Apple Music, etc. Basically anything that streams audio that I've used pauses when I get a Snapchat notification. And if it's "So-and-so is typing" and then "Message from so-and-so" it stops twice! So annoying. But not Spotify's problem.
Also, none of the proposed fixes in this thread have worked for me permanently.
Here’s the solution! Clear your snapchat cache, it worked perfectly for me. Go into Settings and scroll all the way down to Account Actions, right under it says clear cache. Click it and follow the instructions it gives you.