2 added 471 characters in body
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Thread[Plus[Thread[foo[{a, b}, {c, d}]]
foo @@@ Thread[{{a, b}, {c, d}}]
foo @@@ Transpose[{{a, b}, {c, d}}]
Inner[foo, {a, b}, {c, d}, List]
MapThread[foo, {{a, b}, {c, d}}]

{foo[a, c], foo[b, d]}

For Plus you can simply use (thanks: MichaelE2)

{a, b} + {c, d} (* or *)
Plus[{a, b}, {c, d}] 

to get

{a + c, b + d}

Actually, the same is true for all Listable functions such as Times and Power:

Times[{a, b, c}, {x, y, z}]

{a x, b y, c z}

{a,b,c}^{x,y,z}

{a^x, b^y, c^z}

Thread[Plus[{a, b}, {c, d}]]

{a + c, b + d}

Thread[foo[{a, b}, {c, d}]]
foo @@@ Thread[{{a, b}, {c, d}}]
foo @@@ Transpose[{{a, b}, {c, d}}]
Inner[foo, {a, b}, {c, d}, List]
MapThread[foo, {{a, b}, {c, d}}]

{foo[a, c], foo[b, d]}

For Plus you can simply use (thanks: MichaelE2)

{a, b} + {c, d} (* or *)
Plus[{a, b}, {c, d}] 

to get

{a + c, b + d}

Actually, the same is true for all Listable functions such as Times and Power:

Times[{a, b, c}, {x, y, z}]

{a x, b y, c z}

{a,b,c}^{x,y,z}

{a^x, b^y, c^z}

1
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Thread[Plus[{a, b}, {c, d}]]

{a + c, b + d}