I have known many immediate relatives of the U.S citizens and they have never had a problem. However, theoretically the problem exists. It would be right to do an immigration pre-clearance somewhere close to home.
Regarding your query on medical emergencies, you could tell them that your parents took the medical treatment. As long as there was no government funding I do not see any problem.
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Note: Unless the context shows otherwise, all answers here were provided by Rajiv and were compiled and reported by our editorial team from comments, blog and community calls on immigration.com. Where transcribed from audio/video, a verbatim transcript is provided. Therefore, it may not conform to the written grammatical or syntactical form.
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