Yes, in Japan parents and children bath together fully naked. And that's culturally perfectly normal. From a Japanese perspective, together tub-time is good for family bonding.
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From a Japanese perspective, together tub-time is good for family bonding. As children grow older, they'll start enjoying bath time separately. But the habit of sharing the splash can go till junior high or even high school. Japan has a long tradition of communal bathing with onsen and sento.
Couple onsens are special spaces where the two of you would never be disturbed by other people. Private onsens in guest rooms, in particular, let you spend time with your sweetheart enjoying the hot spring all day long, without the need to go out the door. The experience is largely different from normal dates.
No one knows the reason why, but seem to guess that it's to save money in the public restrooms (both the cost of soap and of water). And a lot of Japanese homes have (or used to have) the toilets with the faucet on top and (for some reason) you wouldn't want the hand soap to run back into the tank.
This social aspect, however, led many onsen and sento to prohibit tattooed guests. The Japanese taboo toward tattoos stems from their association with members of Japanese organized crime. Gangsters in Japan are typically heavily tattooed, and body art in Japan came to be associated with unsavory characters.