Is there a connection betweem "a parting shot at the end of a discussion" and the Parthian horse archer practice of wheeling from the battle line and firing an arrow on the run?
(Sorry ’bout the wall of text, but this seems like a complicated one!)
tl;dr version: this seems like a sort of convergent evolution, two phrases coming from different roots but ending up close enough that they effectively gravitated together and merged as a single cliché. The key point: parting and Parthian each have earlier uses, quite distinct from each other, which convincingly lead up to parting/Parthian shot respectively.
The OED cites the figurative use of Parthian going back to c.1640, in the sense of fighting somewhat dishonestly, and in particular of attacking while retreating:
…To look upon this Parthian Fight / Of Love, still flying, or in chase, / Never encountering face to face. —E. Waller, Phillis, 1640.
But the association of this style with the Parthians predates the figurative use:
Or like the Parthian I shall flying fight. —Shakespeare, Cymbeline, 1518.
On the other hand, parting kiss and parting blow first appear in ?1570 and 1592 respectively, the latter used much like parting shot:
Thus much I must say for a parting blow. — R. Greene, Quip for Vpstart Courtier. 1592.
Similar uses of parting continue through the 17th and 18th centuries; the earliest use of Parthian they give that directly parallels it is in 1842:
They have probably enough dealt a Parthian shot to British interests, by setting the Nacional once more upon its legs. — The Times, April 20, 1842.
So it appears that parting blow was used in this sense significantly earlier; but also that Parthian was acquiring these associations independently, in e.g. the Cymbeline quote, well before the wording had converged enough that they would be likely to have influenced each other. But it looks to me as though the existing parting blow/shot/shaft/etc. might well have influenced the later appearances of Parthian blow, etc.
However, the OED’s etymologers themselves suggest an influence in the opposite direction, saying circumspectly (under the etymology of parting):
“In parting shot (see Compounds 1a) perhaps partly influenced by Parthian shot at Parthian adj. 2.”
As Cubbi’s and Mr. S & N’s references show, though, it’s possible to make arguments for influences in either direction, or for none at all.
Dictionaries seem to agree:
American Heritage Dictionary of Idioms: "This idiom apparently originated as a corruption of Parthian shot"
Collins English Dictionary: "Also called: Parthian shot"
Dictionary.com Unabridged based on Random House: "perhaps by folk etymology from Parthian shot"
Also, this post on phrases.org.uk gives a quote from "Fighting Words: From War, Rebellion, and other Combative Capers", saying "Most writers believe that this expression is a distortion of 'Parthian shot,' although there is no firm evidence to support this etymology"
Wikipedia says that these two terms are distinct and unrelated. The Corpus of Historical American English shows no references to "parthian shot" before 1880 but does show "parting shot" before that.