We consider the following class of fractional parametric problems lefttrue(−ΔDirtrue)su=f(x,u)+tφ1+hleft4.ptin4.ptnormalΩ,leftu=0left4.pton4.pt∂normalΩ,where Ω⊂double-struckRN is a smooth bounded domain, s∈(0,1), N>2s, true(−ΔDirtrue)s is the fractional Dirichlet Laplacian, f:normalΩ¯×R→R is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti–Prodi type assumptions, t∈R, φ1 is the first eigenfunction of the Laplacian with homogenous boundary conditions, and h:Ω→R is a bounded function. Using variational methods, we prove that there exists a t0∈R such that the above problem admits at least two distinct solutions for any t≤t0. We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem.