This has nothing to do with the proportion of the genome that is coding as suggested in another answers.
The reason the figures are so different is because they are measuring different things. The chimp-human figure is measuring sequence similarity whilst the figure for kin are measuring gene similarity by descent.
This gives two sources of difference in the two numbers: firstly, there is the distinction between sequence similarity and gene similarity. Let us suppose you have a 1000bp gene, which contains or does not contain a single nucleotide polymorphism. These two variants are different alleles so if you're considering gene similarity then you would say these are 100% different but if you're comparing sequence similarity then you would say that there is just a 1bp difference so these are 99.9% similar. This is how chimps can be both so similar to humans and so different - it is not a matter of how many base pairs are different but where those differences are. A single base pair change in a 1000 bp gene can have a dramatic effect.
Secondly, the figure for relatedness to your parents/siblings is talking about similarity by descent. In fact, your actual gene similarity to your parents is likely to be substantially higher than 50% because your parents are likely to have the same version of many genes so which parent you inherit the gene from makes no difference. Additionally, they will also be homozygous for a number of alleles so that you always inherit the same version of this gene from that parent and are thus more related to your siblings than a crude calculation of relatedness would suggest. The 50% figure comes from only considering two genes the same if they come from the same source, i.e. similarity by descent.
Another, perhaps, better way of thinking of this would be: among those genes that exist in different forms in your parents you will share 50% of them with your parents and 50% of them with each of your siblings on average.
As a bit of an aside: the figure for similarity between human and chimp DNA you've given is based on crude annealing. So if a highly similar piece of DNA occurs in both human and chimp DNA they will match up but this does not consider how the DNA is arranged within the genome so it actually misses a considerable amount of the variation between organisms.