Why do some members of a household specialize in the labor market and other household members specialize in the household sector?
When economists speak of the "household sector" they generally are contrasting it with the "production sector." The household sector provides labor to the production sector and buys goods and services from it. The production sector buys the labor of the household sector and sells goods and services to it. In this model, there is no difference between the "household sector" and the "labor sector."
I assume, then, that you are asking why some people work at home while others work in the formal labor force. If that is the case, let us look at two factors.
First, there is the factor of human resources. The workers who are most valuable to firms are those who have the skills those firms need. These skills can be gained through education and training or through work experience. If people have not been able to get the required skills, they may not be employable in what you are calling the "labor sector" and may therefore have no choice but to work in the "household sector."
Second, there are cultural factors at play. There are cultures in which it is expected that women will remain in the household sector. This can mean that women do not get the education and training needed for paid employment. Alternatively, it can mean that women get the qualifications needed but are still not hired for positions that they are qualified for. In addition, some may make the personal choice to work in what you call the household sector.
There is no one reason for this question. I would say that there are three important variable to consider.
First, we have to deal with the natural talents and desires of people. Some people like to work in the labor market, such as in banking, teaching, or in something else. Others like to cultivate a career or life in the household sector. I would say that this desire is one of the most important points to keep in mind.
Second, there is also the factor of what society needs. Even if a person has desires, he or she might not be able to go in that direction if there are no openings. So, we need to keep in mind the needs of society at a given time. For example, there is almost a 9% unemployment rate today. Jobs are hard to come by.
Finally, there is a matter of timing. Sometimes it really boils down to timing. If the timing is not right, then a person may not be able to pursue something.