The standard definition of an odd function in multiple variables is f(-x, -y) = -f(x, y). Some functions can be categorized as odd or even only in a specific set of variables while being parametrized by others. For example, a family of even functions of x and y, parametrized by z, can be expressed as F(x, y; z) where F(-x, -y; z) = F(x, y; z) holds true for all values of z. It's important to specify the variables when discussing evenness or oddness to avoid confusion. Overall, clarity in variable specification is crucial in defining the properties of multivariable functions.