All known nontrivial parametric solutions of the diophantine equation x₁⁴ + x₂⁴ + x₃⁴ + x₄⁴ = y² give the values of x_i, i = 1, ..., 4, in terms of polynomials in two parameters. In the three simplest solutions, these polynomials are of degrees 2, 4 and 12 respectively. In this paper we obtain infinitely many solutions of the equation under consideration in terms of quadratic polynomials in two parameters, as well as a solution in terms of quartic polynomials in three parameters. We also show how more general multi-parameter solutions of the equation may be obtained.