Mon père préfère les pommes de terre aux haricots, mais ma sœur aime les deux.

Because French possessive adjectives agree with the thing possessed, not with the owner. Père is a masculine singular noun, so you use mon. Sœur is a feminine singular noun, so you use ma. So:

mon père = my father
ma sœur = my sister

In French, when you talk about things in general, especially with verbs like aimer and préférer, you usually use the definite article. So:

les pommes de terre = potatoes, in a general sense
les haricots = beans, in a general sense

English often drops the article here, but French usually does not.

That is simply the normal French expression for potatoes. Literally, it means apples of the earth/ground, but you should learn it as one fixed expression. The singular is:

une pomme de terre = a potato

The plural is:

des pommes de terre or les pommes de terre = potatoes

The verb préférer is normally built with à when you compare two things:

préférer A à B = to prefer A to B

So:

Mon père préfère les pommes de terre aux haricots
= My father prefers potatoes to beans

French does not use que in this structure.

Because aux is the required contraction of à + les.

So:

à + les = aux

That means:

aux haricots = to the beans / over beans, depending on context

After préférer, it gives the second item in the comparison.

In this expression, pomme is the main noun, and de terre describes what kind of pomme it is. So only the main noun changes for number:

une pomme de terre
des pommes de terre

The word terre stays singular in this fixed expression.

Here, les deux means both of them, referring to the two foods already mentioned: les pommes de terre and les haricots.

So:

ma sœur aime les deux = my sister likes both

French often uses les deux where English would simply say both.

Because deux by itself usually just means the number two. Les deux points to two specific things already mentioned and means both or the two.

So:

deux = two
les deux = both / the two ones we are talking about

Because aimer can mean both to like and to love, depending on context. With food, hobbies, and general preferences, it usually means to like.

So in this sentence:

ma sœur aime les deux
most naturally means
my sister likes both

Yes, a few common ones:

père: the è gives an open eh sound
sœur: œ is a special French vowel sound, not exactly like English
haricots: the final s is silent
deux: the final x is silent

A rough pronunciation of les deux is lay duh.