In my experience, there are at least two different approaches to parents playing games with kids, related to two different views of what children are: Pets who can talk or small human beings who don't yet know much. The former provides a justification for the adult cheating against himself, deliberately playing badly in order to give the child a chance. The latter implies that children, like other people, are entitled to honest treatment, and pretending to try to win when you are actually trying to lose does not qualify.
The simplest version of the second approach, and the one I am familiar with from my own experience, is the sliding handicap. Our house had a basement with a ping-pong table, and I spent a good deal of time playing ping-pong with my father. The rules were very simple. I started with some number of points, and whichever of us got to 21 points first won. Every time I won, my starting number went down by one, making the next win harder. Every time I lost, my starting number went up by one, making the next win easier. The result was that the typical game was close, decided by how well each of us played—a more interesting interaction than if we had played without a handicap and my father, who for most of the relevant period was a better player than I was, had deliberately thrown some of the games in order to "make it fun" for me. And the sliding handicap provided a longer run metagame as well, in which my objective was to push the handicap down as far as possible—ideally, in the sufficiently long run, to zero and below.
The approach can be applied, and no doubt is, to a wide variety of other games, as when the better chess player spots his opponent a piece by removing it at the beginning of the game.
What about a game that, unlike ping-pong, is asymmetric, and as a result easier for one side than the other to win? Consider, for example, a board game based on the battle of Gettysburg. The two armies in that battle were quite different, as were their objectives. Unless the designer of the game makes a point of tuning the rules to make victory equally easy for either side—which, of course, he might do—one would expect one side to start with an advantage. The same could be the case for a more abstract game, such as one of the variants of Tafl, a family of early European games of which the best recorded example is Tablut, discovered and recorded by Linnaeus during his travels in 18th century Finland.
In the Tafl games, one side represents a king and his defenders, starting in the central portion of the board. The other represents the attackers, starting around the periphery. The objective of the attackers is to kill the king, the objective of the king is to escape the board. Not surprisingly, in most of the variants, which differ mainly in the size of the board and the number of pieces, one objective is easier to accomplish than the other.
The problem with an asymmetric game is that the handicap doesn't slide. It works fine for two unequal players who are going to stay about equally unequal, but not for the parent/child situation where the child will, with luck, be gradually catching up to the parent. Are there examples of asymmetric games that solve that problem, perhaps by a range of starting scenarios of increasing difficulty for one side, decreasing for the other? The obvious ones are computer games where the player can set the difficulty level against the computer—are there good two player games that work that way?
I cannot resist the temptation to end this post, more random in its subject matter than most of mine, with a quote from the page on Tafl that I earlier linked to:
"Evidence shows that the game of Tablut, described by a traveller called Linnaeus during his trip to Finland in 1732 ..."
Presumably the author of that comment knows more about the history of games than the history of biology.