Paraphyses are part of the fertile spore-bearing layer in certain fungi. More specifically, paraphyses are sterile filamentous hyphal end cells composing part of the hymenium

Hymenium

The hymenium is the tissue layer on the hymenophore of a fungal fruiting body where the cells develop into basidia or asci, which produce spores. In some species all of the cells of the hymenium develop into basidia or asci, while in others some cells develop into sterile cells called cystidia or...

 of Ascomycota

Ascomycota

The Ascomycota are a Division/Phylum of the kingdom Fungi, and subkingdom Dikarya. Its members are commonly known as the Sac fungi. They are the largest phylum of Fungi, with over 64,000 species...

 and Basidiomycota

Basidiomycota

Basidiomycota is one of two large phyla that, together with the Ascomycota, comprise the subkingdom Dikarya within the Kingdom Fungi...

 interspersed among either the asci

Ascus

An ascus is the sexual spore-bearing cell produced in ascomycete fungi. On average, asci normally contain eight ascospores, produced by a meiotic cell division followed, in most species, by a mitotic cell division. However, asci in some genera or species can number one , two, four, or multiples...

 or basidia respectively, and not sufficiently differentiated into specialized, swollen, often protruding cells to be called cystidia.

Paraphyses can also be found in ferns and mosses referring to the filament-like structure that is associated with the fertile sporangia.