Paul has refused to accept gifts from the Corinthian church in order to avoid a patron/client relationship. Rather than patronage, Paul describes his relationship with Corinth as a parent and child (12:14b-15).
The children are not required to “save up” for their parents. This is not the daily needs of children, but rather the family responsibility for building wealth to pass down to the children. In a Greco-Roman context, family name and wealth was extremely important. The father was responsible for creating wealth and prestige for the family and the family name. This wealth and prestige could then be passed down to the children with they were mature enough to contribute to the honor of the family.
In fact, children were not able to contribute to the family honor until they were mature. Paul may be implying the role of the Corinthian church is to grow in maturity themselves! This is consistent with how Paul’s correspondence with the Corinthians began, they were still immature and not able to move on to the “deeper things.” In 1 Cor 4:14-15 Paul describes himself as a father to the church, even if thy have many other teachers.
Within the metaphor, Paul is the patron/father figure who is doing everything he can to create an environment where the Corinthian children can grow to maturity and contribute to the family for themselves. Looking ahead to the next generation, they will “have children” and the church’s responsibility is to maintain and grow the wealth and honor of the family so they too can grow to maturity.
Paul is willing to spend everything he has for the church because he loves them as his dear children. Paul sees his relationship with the church in terms of a family in which he is like a parent and the church are children. He does not want to accept patronage from the church because it changes the relationship and would give the Corinthian church certain privileges over Paul.
This is a powerful image of the relationship of a pastor and congregation. While Paul does want churches to care for the needs of the people who serve in the church, his model for ministry is a caring parent who does everything to give the children want they need to succeed. Pastors who considered the local church their personal kingdom or use the church to enhance their own wealth and prestige have failed to follow the model Paul gives here in 2 Corinthians or the model of humble service demonstrated by Christ.