会話中はネイティヴでもこういう場合meを使いますが。。。両文とも、厳密にいえば「I」が正しいです。
確認の仕方として、文の最後に略された部分を完成させると、正しい形が見えてきます。
My dad was as tired as me. ⇒My dad was as tired as I was. (My dad was as tired as me wasは×)
That girl comes here as often as me. ⇒That girl comes here as often as I do. (That girl comes here as often as me doは×)
口頭ならmeでも自然で問題ありませんが、論文には書きません。
何となく解りました。
benkei