Sandor is right. Indeed, in Hartshorne's example the total family $Y$ is defined by the ideal

$$I=(a^2(x+1)-z^2, ax(x+1)-yz, xz-ay, y^2-x^2(x+1)) \subset k[a, x,y,z],$$ whereas the central fibre (corresponding to $a=0$) is defined by

$$I_0 = ( z^2, yz, xz, y^2-x^2(x+1)) \subset k[x,y,z].$$

The following Macauley2 script shows that

• $I$ is a radical ideal, hence $Y$ is a reduced affine scheme, that is $Y=Y_{red}$.

• $I_0$ is not radical. In fact, the central fiber $Y_0$ has an embedded point at the node $(0,0,0)$ corresponding to the nilpotent element $z$.

  i1 : k=ZZ/32003;

  i2 : S=k[a,x,y,z];

  i3 : I=ideal (a^2*(x+1)-z^2, ax(x+1)-yz, xz-ay, y^2-x^2(x+1));

  o3 : Ideal of S

  i4 : I==radical I

  o4 = true

  i5 : T=k[x,y,z];

  i6 : I0=ideal (z^2, yz, xz, y^2-x^2*(x+1));

  o6 : Ideal of T

  i7 : I0 == radical I0

  o7 = false

Sandor is right. Indeed, in Hartshorne's example the total family $Y$ is defined by the ideal

$$I=(a^2(x+1)-z^2, ax(x+1)-yz, xz-ay, y^2-x^2(x+1)) \subset k[a, x,y,z],$$ whereas the central fibre (corresponding to $a=0$) is defined by

$$I_0 = ( z^2, yz, xz, y^2-x^2(x+1)) \subset k[x,y,z].$$

The following Macauley2 script shows that

• $I$ is a radical ideal, hence $Y$ is a reduced affine scheme, that is $Y=Y_{red}$.

• $I_0$ is not radical. In fact, the central fiber $Y_0$ has an embedded point at the node $(0,0,0)$ corresponding to the nilpotent element $z$.

  i1 : k=ZZ/32003;

  i2 : S=k[a,x,y,z];

  i3 : I=ideal (a^2*(x+1)-z^2, a*x*(x+1)-y*z, x*z-a*y, y^2-x^2*(x+1));

  o3 : Ideal of S

  i4 : I==radical I

  o4 = true

  i5 : T=k[x,y,z];

  i6 : I0=ideal (z^2, y*z, x*z, y^2-x^2*(x+1));

  o6 : Ideal of T

  i7 : I0 == radical I0

  o7 = false