"My father likes playing the piano."
Translation:A mi padre le gusta tocar el piano.
I just wrote out an explanation for this to another similar question, so forgive me for giving you a link to it:
A mi padre is defining the indirect object pronoun 'le' here.
This sentence translates to:
a mi padre - to my father
le gusta - it pleases him
tocar - to play
el piano - the piano
But because - to my father it pleases him to play the piano is a odd grammatical construct, it changes to my father likes...
Without A mi padre, le gusta tocar el piano would mean:
it/he/she/you (usted) pleases to play the piano.
So you need the A mi padre to define what 'le' is referring to.
To play (an instrument) = tocar. Toco el piano.
To play (a game) = jugar. Juego al tenis.