Mathworld gives 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
$$(a^3+2abc)^4(b-c)^4+(b^3+2abc)^4(c-a)^4+(c^3+2abc)^4(a-b)^4=2(ab+ac+bc)^8$$
(equations 144 and 146).
The Ferrari Identity gives
$$(a^2+2ac-2bc-b^2)^4+ (b^2-2ba-2ca-c^2)^4+ (c^2+2cb+2ab-a^2)^4 = 2(a^2+b^2+c^2-ab+bc+ca)^4$$
There is also Ford's Theorem: if $S_j=\sum_{i=j\,(\text{mod}\,3)}(-1)^i\binom ki a^{k-i}b^i$ then $$(S_0-S_1)^4+(S_1-S_2)^4+(S_2-S_0)^4=2(a^2+ab+b^2)^{2k}$$