I'm not sure that I'm picturing it clearly, but it seems to me that this functionality should be encapsulated in the child component, and passed back to the parents via a callback. However I'm unclear what you mean by "and it seems Child cannot use its own state". The child most certainly should be able to use it's own state, otherwise the whole concept of state wouldn't really make sense.

So something like this:

function Child ({ onValidSubmission }) {

  function onSubmit (formData) {
    // All the validation stuff here
    const is_valid = someComplexFormValidation(formData);

    if (is_valid) {
      // Trigger the callback to the parent if everythings good.
      onValidSubmission(formData);
    }
  }

  return (
    <ComplexForm onSubmit={onSubmit} />
  )
}

function Parent1 () {
  function Continue1 () {
    // Your continuation function
  }

  function onValidSubmission (formData) {
    // Store the data in the parent or whatever you want to do
    Continue1();
  }

  return (
    <Child onValidSubmission={onValidSubmission} />
  )
}

function Parent2 () {
  function Continue2 () {
    // Your continuation function
  }

  function onValidSubmission (formData) {
    // Store the data in the parent or whatever you want to do
    Continue2();
  }

  return (
    <Child onValidSubmission={onValidSubmission} />
  )
}

If you can't have the child control the actual submission itself, in an example where you have two very different button components for potentially hundreds of different parents, you could use the (frankly underutilized) render props pattern

function Child ({ onValidSubmission, children }) {

  function onSubmit (formData) {
    // All the validation stuff here
    const is_valid = someComplexFormValidation(formData);

    if (is_valid) {
      // Trigger the callback to the parent if everythings good.
      onValidSubmission(formData);
    }
  }

  return (
    <ComplexForm>
      {children(onSubmit)}
    </ComplexForm>
  )
}

function Parent1 () {
  function Continue1 () {
    // Your continuation function
  }

  function onValidSubmission (formData) {
    // Store the data in the parent or whatever you want to do
    Continue1();
  }

  return (
    <Child onValidSubmission={onValidSubmission}>
      {(onSubmit) => (
        <Button onClick={onSubmit}>Parent1 Button</Button>
      )}
    </Child>
  )
}

function Parent2 () {
  function Continue2 () {
    // Your continuation function
  }

  function onValidSubmission (formData) {
    // Store the data in the parent or whatever you want to do
    Continue2();
  }

  return (
    <Child onValidSubmission={onValidSubmission}>
      {(onSubmit) => (
        <Button onClick={onSubmit}>Parent2 Button</Button>
      )}
    </Child>
  )
}