Well, you never actually calculated the tangent, so it is not that surprising you didn't get the correct result.
t = linspace(-2.5,2.5,100);
x = t.^3 - 4*t;
y = t.^2;
dydt = 2*t;
dxdt = 3*t.^2 - 4;
dydx = dydt./dxdt;
i=1:10:100;
ts=t(i);
xs=ones(size(i));
ys=dydx(i);
tx = ts.^3 -4*ts;
ty = ts.^2;
figure
hold on
plot(x,y,'LineWidth',2)
quiver(tx,ty,xs,ys,'LineWidth',2,'Color','r')
axis equal
Apologies for rearranging your code to my liking. See this page for the math of calculating the tangent of a parametric equation. (You'll need to be careful if dx/dt is ever zero, because you divide by that.)
Also, I was lazy and just set the length of the x-component of the tangent vector to (positive) 1, then calculated the y-component. You probably want something less lazy.
