how to fix a child form in a parent form at a fixed position?

You certainly can add a Form as a 'child form' of another Form, and it's draggable "as is," as long as you expose the TitleBar:

private void Form1_Load(object sender, EventArgs e)
{
    Form2 f2 = new Form2();
    f2.TopLevel = false;
    //
    f2.Parent = this;
    //
    f2.ControlBox = false;
    f2.MinimizeBox = false;
    f2.MaximizeBox = false;
    f2.ShowInTaskbar = false;
    f2.FormBorderStyle = FormBorderStyle.FixedToolWindow;
    f2.Text = "Form2";
    f2.Show();
}

But, note that in this scenario we are not setting the second Form to be an MDIChild of the first Form which we have defined as an MDIParent Form: so in this case setting the FormStartPosition of the second Form by something like:

f2.FormStartPosition = FormStartPosition.CenterParent; 

Will have no effect.

In general, both using MDI architecture, and using Forms within Forms is not that good an idea, and MDI is now "deprecated," meaning not considered a good, modern, architecture.

Please think about making the second Form 'owned' by the first Form:

private void Form1_Load(object sender, EventArgs e)
{
    Form2 f2 = new Form2();
    f2.TopLevel = true;
    //
    f2.Owner = this;
    //
    f2.StartPosition = FormStartPosition.CenterScreen;
    f2.ControlBox = false;
    f2.MinimizeBox = false;
    f2.MaximizeBox = false;
    f2.ShowInTaskbar = false;
    f2.FormBorderStyle = FormBorderStyle.FixedToolWindow;
    f2.SizeGripStyle = SizeGripStyle.Hide;
    f2.Text = "Form2";
    f2.Show();
}

And see how this works for you. You can easily write a 'Move event handler for the second Form to keep it inside the bounds of the first Form: if that's important to your design, or, if you enable Form2 to be resized you can handle the ReSize event and, if you wish, make Form2 appear "within" Form1.