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In this paper, we propose a new reciprocal-orthogonal parametric discrete Fourier transform (DFT) by appropriately replacing some specific twiddle factors in the kernel of the classical DFT by independent parameters that can be chosen arbitrarily from the complex plane. A new class of parametric unitary transforms can be obtained from the proposed transform by choosing all its independent parameters from the unit circle. One of the special cases of this class is then exploited for developing a new one-parameter involutory discrete Hartley transform (DHT). The proposed parametric DFT and DHT can be computed using the existing fast algorithms of the DFT and DHT, respectively, with computational complexities similar to those of the latter. Indeed, for some special cases, the proposed transforms require less number of operations. In view of the fact that the transforms of small sizes are used in some image and video compression techniques and employed as building blocks for larger size transform algorithms, we develop new algorithms for the proposed small-size transforms. The proposed parametric DFT and DHT, in view of the introduction of the independent parameters, offer more flexibility in achieving better performance compared to the classical DFT and DHT. As examples of possible applications of the proposed transforms, image compression, Wiener filtering, and spectral analysis are considered.