Loss-of-function mutations in 3'-to-5' exoribonucleases have been implicated in hereditary human diseases. For example, PARN mutations cause a severe form of dyskeratosis congenita (DC), wherein PARN deficiency leads to human telomerase RNA instability. Since the DC phenotype in PARN patients is even more severe than that of loss-of-function alleles in telomerase components, we hypothesized that PARN would also be required for the stability of other RNAs. Here, we show that PARN depletion reduces the levels of abundant human Y RNAs, which might contribute to the severe phenotype of DC observed in patients. Depletion of PAPD5 or the cytoplasmic exonuclease DIS3L rescues the effect of PARN depletion on Y RNA levels, suggesting that PARN stabilizes Y RNAs by removing oligoadenylated tails added by PAPD5, which would otherwise recruit DIS3L for Y RNA degradation. Through deep sequencing of 3' ends, we provide evidence that PARN can also deadenylate the U6 and RMRP RNAs without affecting their levels. Moreover, we observed widespread posttranscriptional oligoadenylation, uridylation, and guanylation of U6 and Y RNA 3' ends, suggesting that in mammalian cells, the formation of a 3' end for noncoding RNAs can be a complex process governed by the activities of various 3'-end polymerases and exonucleases.