Just to be clear: since Reactions (like Dodge and Parry) can't be used during a character's own Turn (see description for Reactions on page 236), does this mean that if a character swings at an enemy, and that enemy makes a successful Parry followed with a successful Counter Attack, the character that made the 'provoking' attack is unable to Dodge or Parry the Counter Attack?

The RAW is confusing. While page 236 does say that a reaction can only be used when it isn't your turn, there are talents like Furious Assault which only work if you do spend the reaction on your turn.

I'd suggest allowing counter attack to be dodged and parried. But I'm interested in seeing what FFG say.

I allow it (have since DH) - it leads to very amusing, exciting melee exchanges: attack - parry - couterattack - parry - counterattack - dodge (with step aside) and so on. Players can see that all those points invested into their characters work!

Good point....
It might play test it next RT session, might be interesting....though this would make the Vindicare Melee Emperor...