This sentence, while 100% grammatical, sounds kind of awkward, and I am sure that, as a fellow Japanese-speaker, you probably felt the same way.
Even though it is a little awkward, it probably could not be improved much because it has been written to express something that the Japanese language is not designed well to express.
It is awkward because it is written by the vendors (adults) to the customers (children) and in what is essentially supposed to be an honorific language. Imagine having to write instructions using keigo to someone who cannot even read kanji!
(I sometimes hear the same kind of awkwardness/unnaturalness in restaurants, shops, etc. in Japan where the adult employees must sometimes speak to kids. I would not do any better than they do, either, if I worked at one of those places.)
This instruction uses 「かた」 to refer to the child customer, and I would say that it is a reasonable word choice. The manufacturers, however, used 「人」 to refer to the kid's parents when the parents themselves are likely to be the ones who actually payed for the product. My guess is that the instruction-writer tried to see things from the kid's viewpoint and ended up using 「おうちの人」 as that might sound "better" or "more natural or familiar" to the kid him/herself.
(Kids are generally pretty poor in their keigo usage. The manufacturers know that, but they could not completely ignore keigo since the kids are their customers and customers are gods in Japan. This dilemma, according to me, helped create the awkwardness of the instructions.)
In conclusion, I find it difficult to improve the sentence. One could use 「かた」 twice, but my bigger concern would be the fact that even if one used 「かた」 twice, it would not improve the awkward and unnatural verb phrases that should go with the polite and respectful 「かた」. I am referring to 「わからない」 and 「みせてください」, both of which are "plain" and would not be used in product instructions for adults, but using 「おわかりにならない」 and 「おみせになっていただくようおねがいいたします」 instead would be out of the question with kids.
would be risky because one could end up sounding so 「上から」 by addressing your customer with 「人」 even if s/he is a kid. It would sound like an instruction on a cheap candy. Just to play safe, I would pesonally not use the sentence in highlight above as an instruction in a game. If you used it, however, not many people would feel it strange, either.
Adult Japanese-speakers would know instinctively that this is not a good situation to use keigo properly or even nicely for that matter.