How do you consider an organism to be a new species?
There are various hypotheses and concepts proposed to define a species, but no definitive answer. Mostly a combination of these concepts are used because each hypothesis has exceptions. Classification is constantly being debated, reevaluated, and changed as we learn more. Some of the most notable definitions are:

The is the biological species concept in which Ernst Mayr defines a species as actually or potentially interbreeding populations that are reproductive isolated from other groups and produce viable, fertile offspring. But this doesn't by definition exclude hybridization between species, but often hybridization does not produce viable or fertile offspring. (This definition doesn't work with parthenogenesis. Also, hybridization is accepted as a mechanism that can lead to speciation, most notably seen in plants, where the hybrid offspring can interbreed with other hybrids, but are not able to reproduce with their parental species.)

There is also the morphological species concept which uses morphology and body features to classify organisms, which is more useful when there is little population information available, such as for fossils. But this can result in one sexually dimorphic species being classified as two. (The eclectus parrot was initially classified as two species because the males are green, while females are red.)

In the phylogenetic species concept Willi Hennig defines a species as the smallest group of individuals sharing one common ancestor and a set of fixed heritable differences to make up one phylogenetic branch. Genomics is often used to define this. There is debate that this method creates "taxonomic inflation" and creates more species than necessary because different populations may be able to interbreed, are not reproductively isolated, but are genetically differentiated enough due to local adaptation.

The ecological species concept defines species by their environmental niche--organisms that look, act, and share similar life history patterns. This definition includes parthenogenic species as a separate species from its non-parthenogenic relatives.

(It becomes much more complicated when trying to classify prokaryotes and viruses because mutation occurs at a much higher rate and many of the above definitions cannot be effective applied.)

Ultimately, yes, there are various accepted species that reproduce only by parthenogenesis (like the mentioned New Mexico whiptail lizard), and parthenogenesis can be a defining factor which separates them from a sexually reproducing counterpart species that is similar in morphology, niche, and evolutionary history.

Information comes from courses and textbooks from my Ecology &amp; Evolutionary Biology major within the last year.