I can help you! Teacher here. It's not a good idea to start a sentence using "even though". Use just "though". Also, it's not a good idea to use "I" in both the phrases because it's understood you're speaking of yourself. And there IS a comma before "I" !! So I strongly suggest this..."Though tired, I agreed to go shopping with my parents ." The suggestion to use "although" is also acceptable.."Although tired, ", but I think just "though" sounds better because you're speaking of just one instance...going shopping. If you were speaking of more instances, you'd use "although", because it has the prefix meaning "all" as in "although" and there's just one thing going on. So, "Though tired, I agreed to go shopping with my parents ." I hope this helps you!
I believe that sentence is grammatically correct. The only problem is punctuation because there is no comma between "tired" and "I." However, the sentence would sound better if you used "although" instead of "even though."