엄마가 동생을 잘 둘봐 달라고 했어요. (Mum asked to take care of 동생)
엄마가 동생을 잘 돌보라고 했어요. (Mum ordered to take care of 동생)
엄마가 동생을 잘 돌봐 달라고 부탁했어요. (Mum asked to take care of 동생)
In terms of sentence structure, there seems to be no difference between 1 and 3, except that in 1 "하다" is used which is "부탁하다" in 3.
Maybe it is the word "달라고"(contains the meaning of "ask") that makes you confused.
A few more examples might be helpful:
그는 (나에게) 책을 읽어 달라고 했다. (asked me to read a book for him)
그느 (나에게) 밥을 사달라고 했다. (asked me to buy food for him)
그는 (나에게) 가방을 들어달라고 했다. (asked me to carry bag for him)
BTW, "달라고" is more widely used in North Korean than in South Korean.