Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that their solution preserves for all time a uniform pointwise bound in absolute value imposed by the initial and boundary conditions. It has been a challenging problem on how to design unconditionally MBP-preserving time stepping schemes for these equations, especially the ones with order greater than one. We combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and successfully derive the sufficient conditions for the proposed method to preserve MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many strong stability preserving sIFRK (SSP-sIFRK) schemes do not satisfy these conditions, except the first-order one. Various numerical experiments are also carried out to demonstrate the performance of the proposed method.