Okay, I think I’ve found a solution. 
Instead of overloading the parallel operator \parallel to return something non-Boolean, I’ve realized that the notation I seek exists so I’ve chosen to call my harmonic sum +_p where p=-1.
This is similar to the notation for S_p sum of powers, L_p norms, and M_p power means, in which cases p is used to signify the power that the arguments are raised to before performing the summing function. They’re defined as:
S_p(x_1,\ldots,x_n)=\sum_{i=1}^nx_i^p,
L_p(x_1, \ldots, x_n) = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}, and
M_p(x_1, \ldots, x_n) = \left(\frac{1}{n}\sum_{i=1}^n x_i^p\right)^{1/p}.
Similarly I can define
a+_pb=(a^p+b^p)^{1/p},
which is similar to the S_p sum of powers (but taking the result to the 1/p), the L_p norm (but without taking the absolute power), and the M_p mean (but without dividing by n^{1/p}). They could be related as:
S_p(x_1,\ldots,x_n)=(x_1+_p\ldots+_px_n)^p,
L_p(x_1,\ldots,x_n)=(|x_1|+_p\ldots+_p|x_n|), and
M_p(x_1,\ldots,x_n)=\left(\frac{1}{n}\right)^{1/p}(x_1+_p\ldots+_px_n).
For example, the quadratic sum of a and b is a+_2 b = L_2(a,b), and for my harmonic sum I can write:
+₋₁(a::Number,b::Number) = 1/(1/a+1/b)
(Super yay for allowing Unicode characters in operator names! 
)
Fun sidenote, the distributive property a(b+_pc)=ab+_pac holds for any p\ne 0.
To incorporate the implicit meaning of a harmonic sum where two rates add in harmony, I can refer to +_{-1} as simply +_h. So,
+ₕ(a::Number,b::Number) = 1/(1/a+1/b)
This notation is consistent with that for the Kolmogorov-Nagumo generalized f-mean M_f, which is defined for some continuous monotone function f as:
M_f(x_1,\ldots,x_n)=f^{-1}\left(\frac{1}{n}\sum_{i=1}^nf(x_i)\right).
I can write a generalized f-sum as
a+_fb = f^{-1}(f(a)+f(b)).
M_h is the harmonic mean where h(x)=x^{-1}, and with my notation, +_h is the harmonic sum. Similarly, M_q is the quadratic mean and +_q is the quadratic sum where q(x)=x^2.