In this paper we establish a restriction estimate for a class of oscillatory integral operators along a paraboloid, \t\t\tPd−1:={(x1,…,xd):xd=x12+⋯+xd−12}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {\\mathbb{P}^{d-1}:=\\bigl\\{ (x_{1},\\ldots ,x_{d}):x_{d}=x_{1}^{2}+ \\cdots +x_{d-1}^{2}\\bigr\\} .} $$\\end{document} Specifically, we consider the oscillatory integral operators defined by \t\t\t1Tm,nf(x)=∫Rdei(x1mξ1n+⋯+xdmξdn)f(ξ)dξ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ T_{m,n}f(x)= \\int _{\\mathbb{R}^{d}}e^{i(x_{1}^{m} \\xi _{1}^{n}+\\cdots +x _{d}^{m}\\xi _{d}^{n})}f(\\xi )\\,d\\xi , $$\\end{document} where n, m are integers satisfying 2leq d< nleq 2md, then \t\t\t∥Tm,nf∥L2(dσ,Pd−1∩Bd(0,1))≤Cm,n,d∥f∥Lp(Rd)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Vert T_{m,n}f \\Vert _{L^{2} (d\\sigma , \\mathbb{P}^{d-1}\\cap B^{d}(0,1) )} \\leq C_{m,n,d} \\Vert f \\Vert _{L^{p}(\\mathbb{R}^{d})} $$\\end{document} holds for 1< pleq frac{4md}{4md-n}. A necessary condition is also given to ensure this boundedness.