You don’t say enough here to know how much power the new parent currently has with respect to your organization.  For them to change their bylaws to say they can approve all your directors may be notice to themselves of what they consider their power, but changing their bylaws doesn’t change your bylaws to give them such power.

If the new parent is the sole member of your corporation, it probably has the ultimate power to appoint and remove the entire board.  Faced with opposition from directors of a subsidiary, the sole member will be tempted to step in to remove the recalcitrants and proceed with what they want to do.

Even if it is not the sole member and only has the power to appoint a majority of the board, it may be able with that vote to change the bylaws any way it wants.  If it requires a greater percentage to amend the bylaws, the parent may be able to pick off a few of the “independent” directors to get the vote it needs.  If it have the power to approve each new independent director, it may withhold its approval from anyone who won’t agree to make the changes it wants.

In the meantime, the directors of the subsidiary have a fiduciary duty to act in the best interests of the subsidiary, not in the interest of the new parent corporation.  (See Nonprofit Issues®, State Can’t Force Foundation to Contribute for State Programs, 1/16/05.)  You are totally within your right — and have an obligation — to oppose its control over the selection of the independent directors if you think it is in the best interest of your corporation.  It is probably not realistic to expect that the independent directors will stand up to the pressure that the new parent can impose, however, and ultimately the new parent will prevail. 

But that’s why this affiliation was structured so that the new parent has the power to appoint the majority of the board of the subsidiary.  Your organization is part of a new system, not a free-standing, independent organization.