http://mathworld.wolfram.com/DiophantineEquation4thPowers.html equations 144 and 146 give two parametrizations of Ramanujan: if $a+b+c=0$ then

$$a^4(b-c)^4+ b^4(c-a)^4+ c^4(a-b)^4= 2(ab+bc+ca)^4$$

and

$$(a^3+2abc)^4(b-c)^4+(b^3+2abc)^4(c-a)^4+(c^3+2abc)^4(a-b)^4=2(ab+ac+bc)^8$$

And http://mathworld.wolfram.com/FerrarisIdentity.html gives the Ferrari identity

$$(a^2+2ac-2bc-b^2)^4+ (b^2-2ba-2ca-c^2)^4+ (c^2+2cb+2ab-a^2)^4 = 2(a^2+b^2+c^2-ab+bc+ca)^4$$

Also, http://mathworld.wolfram.com/FordsTheorem.html gives Ford's thereom: if  $$S_j=\sum_{i=j\,(\text{mod}\,3)}(-1)^i\binom ki a^{k-i}b^i$$   
then
$$(S_0-S_1)^4+(S_1-S_2)^4+(S_2-S_0)^4=2(a^2+ab+b^2)^{2k}$$