A clue to the possible source/origin of the expression might also be this:
https://artsandculture.google.com/asset/trifacial-trinity-anonymous-cusco-school/nwGOfzqyUQzmpQ
...the Viceregal church sought to explain to the faithful the mystery of the Holy Trinity, so crucial to Christian doctrine. In this case, the image has medieval origins, transmitted through engravings. The Catholic dogma which affirms the existence of three different persons in a single God is expressed here as a face with four eyes, three noses and three mouths.

El dogma católico que afirma la existencia de tres personas distintas en un solo Dios, toma aquí la forma de un rostro con cuatro ojos, tres narices y tres bocas.
The word "narices" has also another variant namely "napias" which means "big nose/s"
"QUÉ SIGNIFICA NAPIAS EN ESPAÑOL
En el diccionario castellano napias significa narices, órgano de la cara humana, especialmente cuando es muy grande."
Source https://educalingo.com/es/dic-es/napias
If "de tres pares de narices" means "the Holy Trinity", then "de tres pares de napias" must mean something bigger than that, or that one received something more than expected.
A search on the internet for the expression "De tres pares de napias" shows 120 hits.