EDIT - Advice... Use royal to start for an easier start. Parry/counter. Keep your shield up. Keep a very measured pace. Try to lure enemies out one at a time to not get swarmed.
I'm near the end of my NG+ and I've never once parried. I just couldn't get the hang of it - every time I tried I would get slaughtered when I missed the parry. So I just went to block/attack with the spear - much easier. I think by parry/counterattack you mean when the enemy attacks you push L2 with the shield on the left hand, right? Anyway, just a tip for those starting out that it is by no means necessary to know how to do that.
Start as Royalty....it will make the first level infinitely easier to finish. Dying is a big part of the game....it is a cruel learning process.
The False Idol on 3-1 has to be the easiest of the bosses I've finished yet...although I hear 4-2 is easy as well, but I have not made it there yet. Finished 1-1, 1-2, 2-1, 2-2, 3-1, and 4-1. Was stupid and upgraded my +6 spear without enough moonstones to get higher than +1...the meat cleaver assisted a little, but I upgraded the Cresent Falchion to +4 and that started kicking major rear....I really miss being able to block and attack though...time to farm 3-1 and 3-2.
Royalty, yes... I only died a few times when I first started out.
3-1 was pretty easy in original game but much harder in NG+ because it seems she made so many more copies and you can't always hide from them (some are in front, some behind even if you hide behind the pillars). Still not too tough, but I had to be much more careful. The black phantom leading up to her, though, was easy as cake on NG+ and I was prepared for her to be much tougher.
4-2 is somewhat easy, but can be a bit tricky if you don't time your runs from the front to the back very well. The essiest boss in the game is the one on 4-3, though - and it is even easier in NG+ because you already have the sword you need to grab from way out on the cliffs (which is the only slightly hard part of the level).
I've stopped playing for now but I need to get back in before I forget how to play effectively. I still need to do the black CT events and then finish my 2nd playthrough (just 1-4 dragon and boss left to do) and then move on to NG++. I need to blow through that and get all my spells, then start NG+++ and get the last few special weapons forged, and I should have a platinum. It will be interesting to see how quickly I can blow through my NG++ when I don't have to do any additional stuff. The problem is, I always have a hard time passing up crystal geckos and like to get to all of them and farm them - not to mention the primevil demons.
A tip for late game on the first playthrough: I would suggest you take on black phantom Satsuki in the original game because even though I am leveled up way beyond what I was when I first fought him, he seems to be much more powerful in NG+ - more so than any of the bosses. I haven't fought any other BF NPC's except Miralda (who I cheesed with poison cloud again) so I'm not sure if they all get tougher like that.