Consider the 3×3 block of live cells:

    .....
    .ooo.
    .ooo.
    .ooo.
    .....

It has no parent with the same symmetry, since every neighbour of the centre cell would be the same but a cell can't live or be born with 0 or 4 neighbours.

But it does have a parent with less symmetry:

    .....
    .....
    ooooo
    .....
    .....