This should be split up into Mr. Spats and Mrs. Spats.
Mr. and Mrs. Spats are the parents of Carmelita Spats.
Nothing is known about Mr. and Mrs. Spat's personalities.
The Spats are briefly mentioned in a letter given to them by Vice Principal Nero. He thanks them for sending him an article from The Daily Punctilio on the dangers of allowing students to read certain books. Nero uses this information to fire a teacher named Ms. K (implied to be Kit Snicket). Ms. K responds by grabbing two students by the ankles and running away with them. Nero assures Mr. and Mrs. Spats that neither of the two children is their darling daughter Carmelita.
In The Slippery Slope, the Man With a Beard But No Hair and the Woman With Hair But No Beard mention all the fortunes they will secure from the Snow Scout's families. One such fortune named is the "Spats fortune."