<image>As shown in the figure, it is known that the vertex of the quadratic function \( y = (x - 1)^{2} \) is \( C \). The graph intersects the line \( y = x + m \) at points \( A \) and \( B \). The coordinates of point \( A \) are \( (3, 4) \), and point \( B \) is on the \( y \)-axis.  
(1) Find the value of \( m \). 
(2) Point \( P \) is a moving point on segment \( AB \) (point \( P \) does not coincide with points \( A \) or \( B \)). Draw a perpendicular line from point \( P \) to the \( x \)-axis, which intersects the graph of the quadratic function at point \( E \). Let the length of segment \( PE \) be \( h \), and the \( x \)-coordinate of point \( P \) be \( x \). Find the functional expression for \( h \) in terms of \( x \), and state the domain of the variable \( x \). 
(3) Under the conditions in (2), let \( D \) be the intersection point of the line \( AB \) and the axis of symmetry of the quadratic function graph. Is there a point \( P \) on segment \( AB \) such that quadrilateral \( DCEP \) is a parallelogram? If such a point \( P \) exists, find its coordinates. If not, explain why.