Nightmare Soul Puppets are equivalent to LESPs in several ways, but much more challenging to acquire.
First, you need to unlock them. Every Nightmare Soul Puppet requires at least 75 normal Soul Puppets to be unlocked along with accomplishing their challenging condition. You'll receive one egg of a Nightmare Soul Puppet upon success.
After that, you can get more via breeding their specific Breeding Combo. Please be warned that creatures in this category have an inherently high chance of failure, both when breeding and when painting.
Trades and auctions with other players are the only other way to get them, but like usual, you will be unable to acquire them this way if you do not have the relevant species unlocked.