- My father's hands → modified by a possessive noun
- The hands of my father → modified by a preposition
When is "my father's hands" preferable over "the hands of my father", and vice versa?
"My father's hands" is nearly always preferable, in America, simply because it is shorter.
Of course you could use "hands of my father" anytime if you like. You would sound formal, or antiquated, or like a non-native speaker—especially if you extended that usage to, for instance "the hands of you" ,"the hands of me", "the hands of him", "the hands of us", "the hands of her", the hands of them". It would simply take you longer to talk.
In Spanish, I believe you have to say it that way. Las manos de mi Padre. Many things take more words to express in Spanish than in English. Apart from colonial history, that might help explain the dominance of English in world commerce. It's quicker.