Yes, 'my' is implied. You can say 'mój tata' if you want, but it's not necessary, unless you want to put an emphasis on 'my' (e.g. when bragging about your father). If you want to talk about someone else's father you need to specify it, e.g. 'twój tata' - 'your dad'.
No, you don't need to use (or assume) 'my'. The sentence works in English without it, when some people refer to their relatives by the name of their relation e.g.
Q. What is Grandma doing? A. Grandma is looking for wolves...
As tadjanow writes, it could also be used to add emphasis.
"The father..." in English sounds like
"The Catholic priest is looking for a good refrigerator." It should just be "Dad..."
Oh dear! Duo needs to learn yet another apostrophised form ("Dad's"), as it just complained I'd "missed a space" by answering "Dad's looking for a good fridge." :~|