The purpose of this paper is to develop a class of generalized parametric Slant-Hadamard transform of order (formula available in paper)where k is an arbitrary integer and to present its fast algorithm. As special cases of this class are the classical Slant-Hadamard (k=2 and β N =1), the generalized Slant-Hadamard (β N =1), and the parametric Slant-Hadamard (k=2) transforms. We will show that the parametric Slant-Hadamard transform is slightly superior to the DCT for compression of the geometric test images at a particular quantization matrix scaling factors.