I just did this recently, The parent VHD has a internal GUID that the child vhd checks, that is why it is failing. If you copy the parent VHD to your other computer and overwrite it (you will need to take ownership of the file as by default only the user SYSTEM has privileges to write to that VHD) it will work fine. 32 and 64 bit does not matter as that is what i did (started on 64 copied to 32, no problems)
One issue have I have encountered is the hibernation state does bomb when you switch computers, but the OS inside the vm just treats it like a power loss an as long as you had no programs running it will be fine.
UPDATE:
bonus points if while you are installing you use the network share of your local computer instead of the C:\ version so you can access the VM over the network, here is a snippet of my VMC file
<ide_adapter>
<ide_controller id="0">
<location id="0">
<drive_type type="integer">1</drive_type>
<pathname>
<absolute type="string">\\schamberlain\vm\Windows XP Mode 1.vhd</absolute>
<relative type="string">.\Windows XP Mode 1.vhd</relative>
</pathname>
<undo_pathname>
<absolute type="string">\\schamberlain\vm\VirtualPCUndo_Windows XP Mode 1_0_0_18514606212011.vud</absolute>
<relative type="string">.\VirtualPCUndo_Windows XP Mode 1_0_0_18514606212011.vud</relative>
</undo_pathname>
</location>
</ide_controller>
<ide_controller id="1">
<location id="0">
<drive_type type="integer">2</drive_type>
<pathname>
<absolute type="string">F</absolute>
<relative type="string" />
</pathname>
</location>
</ide_controller>
</ide_adapter>
When I installed it I used \\schamberlain\vm
as my VM directory.