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You can file multiple H-1B transfers simultaneously, and it's legal. Approval of one transfer doesn’t obligate you to join that employer, nor does it affect your existing H-1B status with your current employer. You can choose to join any employer whose transfer gets approved, provided all transfers were filed while you were in valid status with your current employer.
However, if your H-1B validity is close to expiring (e.g., within 6 months), later-filed transfers may require "bridge petitions" to maintain continuity of status. Consult an immigration lawyer to ensure proper handling and avoid complications.
 

If you are filing an I-130 for your parent, they can generally travel to the U.S. on a visitor visa while the petition is pending. Theoretically, they could face issues at entry due to the pending green card application, but in practice, parents of U.S. citizens are rarely stopped for this reason.
To reduce the risk of being denied entry, consider having your parent use an immigration pre-clearance airport, like Abu Dhabi or Dublin, so they can return home easily if denied. While past practices suggest minimal issues, policies could change, especially under unpredictable administrations.
 

An H-4 visa holder operating a small business in India from the U.S. could potentially violate their visa status, even if earnings are in Indian rupees. At the port of entry, honesty is essential—lying could lead to a permanent U.S. entry ban, which is hard to waive.

While it's unclear if this constitutes a status violation, it’s a borderline case. To avoid risks, it’s advisable not to engage in such activities while on H-4 status.