The equation $x^2+1 = yzt$ has a parametric solution as follows.
We factorize $x^2+1 = yzt$ in $\mathbb Z[i].$
Let $(Y,Z,T)=(a+bi,c+di,e+fi)$ then
$x+i = YZT = (acf+ead+ebc-bdf)i+ace-fad-fbc-bde.$
Hence we get a parametric solution $x = acf+ead+ebc-bdf$ if $ace-fad-fbc-bde =1.$