I'm using Shadow Parasite and Sun Stroke

Response: After you lose a challenge by 4 or more total STR, until theend of the phase, attached character gets -4 STR and is killed if its STR is 0.

If Sun Stroke is attached to a character with strength 1, and I lose a challenge by 4 or more, how much strenght does the character lose ... 1 or 4

Does each Shadow Parasite get +4 strength?

I could control Coldhands (the original one) and attach Sun Stroke to him (so that the attachment stays in place as the character does not get killed)

If I lost two challenges in the one phase, but by 4 or more - does the character get -8 as a result of Sun Stroke ... and thus the parasites get +8

Interesting use of Coldhands, but yes what you surmise is correct. The effect is cumulative and the character is still getting a -4 with Sun Stroke even if their strength is already lower than 4 and each add'l loss by 4 would continue that effect. Each Shadow Parasite can be triggered individually for every strength reduction and will get the accumulated strength of each reduction. So, in the specific (unusual) case where you've lost two challenges by 4 and Coldhands (cannot be killed) has been reduced by 8, each Shadow Parasite would be at 8 STR.

Assuming you're playing a Targ deck (due to the Parasite)... one has to wonder what you're doing to lose two challenges by 4 just to fuel the Parasite however. Sounds like a fun bit of jank though.

Some scenarios may be made up in order to test a principal

That said, I find Sun Stroke useful but its a bit of a pain if I keep killing characters and Sun Stroke gets discarded ... so I was wondering whether there are some alternative uses ... such as with Coldhands

I'm also assuming that I can probably increase my opponents strength to ensure I lose the challenge by 4

Who knows, one day all my plans may come together in an amazing deck!!!