Some times, I come across sentences like - parched, he still went ahead on his ways & being parched, he still went ahead on his ways - but i fail to comprehend as to which one is correct and why it is correct?
I earnestly request you to answer to me asap. Really appreciate your help!
They are both acceptable (although "went ahead on his ways" is rather clunky).
Please note the corrections I have made to your post in red above. Please don't write things like "please answer me asap". We are volunteers and we work at our own pace.
Remember - correct capitalisation, punctuation and spacing make posts much easier to read.