In a Wikipedia article about garden path sentences, parsing is explained:
When reading a sentence, readers will analyze the words and phrases they see and make inferences about the sentence’s grammatical structure and meaning in a process called parsing. Generally, readers will parse the sentence chunks at a time and will try to interpret the meaning of the sentence at each interval. As readers are given more information they make an assumption of the contents and meaning of the whole sentence. With each new portion of the sentence encountered, they will try to make that part make sense with the sentence structures that they have already interpreted and their assumption about the rest of the sentence.
That's what's going on in your sentence:
"I watched them playing with my basketball"
If that means that the people you were watching were playing with your basketball, then the sentence will probably be parsed correctly on the first read. However, if you mean that you were watching the people while you were dribbling your basketball, then there's a good chance that meaning won't be caught on the initial read.
Other answers have already delved into good ways to fix this. You could reorder the thoughts in the sentence, or add additional words to clarify what is going on:
- As I played with my basketball, I watched them.
- I watched them while I played with my basketball.
As for the additional example in your comment ("I watched them using my binoculars"), that could be rephrased, too, to make the intended meaning more apparent:
- I watched them through my binoculars.
- I watched them as they looked through my binoculars.