G-quadruplexes are four-stranded nucleic acid structures found at G-rich regions of the genome, including promoters, origins of replication and telomeres. It is becoming increasingly recognised that G-quadruplexes play specific roles in the regulation of biological processes such as transcription, translation and DNA replication1. Formation of G-quadruplexes is dynamic and regulated by proteins that assist DNA folding or mediate G-quadruplex resolution. Telomeric G-quadruplexes were long believed to form a protective “cap” at telomeres, preventing their extension by the ribonucleoprotein enzyme telomerase. This has led to the development of G-quadruplex stabilising ligands as potential telomerase inhibitors. Contrary to this belief, we have demonstrated that distinct subsets of telomeric G-quadruplexes can be extended by human and ciliate telomerase enzymes2,3. However, a mechanistic understanding of how telomerase deals with structured DNA has remained elusive. Here, using single-molecule fluorescence resonance energy transfer (smFRET) microscopy and telomerase enzymology we provide direct evidence, at single-molecule resolution, that human telomerase can resolve and then extend parallel G-quadruplexes, and we provide a mechanism for this unexpected activity of telomerase. Binding is initiated by the RNA template of telomerase interacting with the G-quadruplex, which maintains a mostly folded structure. With the G-quadruplex correctly aligned, nucleotide addition then proceeds to the end of the RNA template. It is only through the large conformational change of translocation that the G-quadruplex structure is completely unfolded to a linear product. In contrast to prevailing hypotheses, parallel G-quadruplex stabilisation with small molecule ligands did not inhibit G-quadruplex extension by telomerase. These data reveal that telomerase is a parallel G-quadruplex resolvase. The conserved ability of human and ciliate telomerase enzymes to unwind these structures implies that parallel G-quadruplexes form at telomeres in vivo, and do not form a barrier to telomerase extension.