I have never heard of this. It sounds sweet! I wonder if this wood was meant to be burned. Perhaps in the old days when a fireplace would be used to heat the home and cook, to burn the wood when making a meal or warming your family would be good luck. Since this wood was polished and decorative, it probably wasn't meant to be burned, but an old tradition. I don't think I would have burned the wood, but would have kept it and pass it along, like your parents did.
I have always had the superstition that you never move your brooms. It's bad luck to take your broom with you. You leave them behind and buy new ones at the new home. The reason being, you are making a new start and when you bring a broom with you, you are carrying bad events and memories from your old life with you, even if your time in the old location were overall happy. You're not leaving bad luck behind for the new residents, it's just bad luck to take them with you. The new residents may use or dispose of them as they wish. This doesn't apply to mops and vacuums, and I don't know why, but the vacuum would be pretty expensive.
I harbor fewer superstitions than my mom, and her less than her mom, but this is one that I have strictly adhered to.
I remember someone moving to a new location and found it rather odd that there were several brooms left behind. I believe this was a place of business and there were just an unusual number of brooms. I was able to explain this superstition to this person, which was probably the explanation for the brooms. When my husband and I moved, I told him not to move the broom. He did anyway. Huge fight ensued. I'm sure he was just:
My daughter still remembers granny getting upset when she opened an umbrella in the house. She was probably about 2 at the time. I never really stuck to the umbrella rule, so she didn't know. We still get a chuckle at this memory.
My question, if the house doesn't have a fireplace, what do you do? Do you still leave a piece of wood? I think it's a fabulous idea to leave some good luck and well-wishes behind for new residents. If I ever move again, I'm going to consider something like this!
We also always bring salt and bread to a new home. When I moved, I didn't have to buy salt for quite a long time.