It sounds strange without the repeated "play" because you are dealing with two different meanings of "play".
It's like saying "Lefty stole second base and the hearts of the fans."
The reader/listener has to make an unexpected adjustment.
If you don't like "practice guitar", then go ahead and repeat "play" until a better suggestion comes along.
By the way, if your last word were "banjo" instead of "chess", I'd suggest you say "and" twice:
"I like to read and play guitar and banjo."
The problem is that you created a parallel structure with read and play. They are both verbs. Chess is not a verb, so using it as the third item in the list violates the parallel structure. That is why it sounds wrong: because it is. If this were mathematics, you could use parentheses around (guitar and chess) to make it clear that the verb play applies to both of them, but this isn't mathematics. It's English. In English, once you start a parallel structure, you have to stay with it.
Added in edit: The reason the rearrangement in the first list of Copyright's post works is that the "guitar and chess" part comes before a parallel structure with two verbs is created.
I think the main problem is that playing guitar and playing chess are two separate meaning of the word play.
"I like to read and play guitar and cello." would be okay, but personally I'd always separate the reading from the playing. This is what I meant by two meanings of "play". You can play people too, but you'd never write "She likes to play violin and people" unless you were deliberately aiming for confusion on the word.