Are you in status on the H? If so, then you can just send the new I-94. Talk to an attorney, as their visa will have more to do with their immigrant intent than with your visa. I would think your status and support will have very little bearing on their case and they will need to qualify independently. Good luck.
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Neither. Tourist visa requires no sponsorship.
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Not necessary, and even if you did, such "sponsorship" will have very little bearing on parents' ability to obtain visitor visas, since they need to qualify on their own merits by showing both substantial ties to their home country and intent to depart the US after a temporary visit.
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