Must a participial phrase always modify the subject of a sentence, or can it modify the object?
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A participle phrase can modify the subject or the object. For the latter, this page gives the example
The cousin is doing the walking. And for those who deny present participles in English, try
It might help to give an example of what you mean by "participal phrase", but generally speaking the modifiers of a noun are agnostic as to the function of that noun in the overall sentence. So both of these are possible: