The best replacement for custodial parent is non custodial parent.

The best alternatives to custodial parent are rated as following:
★★★★★  non custodial parent
★★★★★  legal guardian
★★★★  residential parent
★★  non-custodial parent
★★  joint custody
★  custodial parent
★  non custodial
★  other parent
  sole custody
  birth parent

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Speaking about custodial parent, one probably shouldn't forget about joint custody.

Other search terms related to custodial parent and their ratings are:
★★★★★ joint custody
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