It's hardly only Russian media has pointed out a connection between the Azov Battalion and neo-Nazism. Even the United States Congress, usually no friend of Russia, cut off funding in 2015 due to these connections. There's a bit of a summary here: https://en.wikipedia.org/wiki/Azov_Battalion#Neo-Nazi_ideolo...
Azov are kind of right-wing (although not outright "nazi"), but this judgement is not (and should not be) based on how their insignia looks. And also, Azov has no connection to Parubiy at all (Azov just reused that unlucky old logo out of the blue).
There is a cultural difference in what Ukrainians and Russians mean when they say "nation". In Ukraine, when you say "nation" you mean basically the same as what a USA president would mean under this word. But in Russia, the word "nation" is considered to be synonymous with "nazi", and is generally avoided.
The leader of the battalion, Biletsky, is also involved in this nice organization: https://en.wikipedia.org/wiki/Social-National_Assembly