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Yes, $A(X) = cX/\sqrt{\log(X)} + O(X/\log^{3/2}(X))$ for a positive real number $c$ which I think is 1. This result holds in much much more generality, e.g. for primes in congruence classes or even for positive-density subsets of primes defined by modular forms. See Theorem 2.8 of Serre's "Divisibilit'e de certaines fonctions arithmetiques" from 1976 in L'Enseignement Mathematique. You'd replace $\sqrt{\log(X)}$ with $\log^{1-\delta}(X)$ for a more general set of primes of density $\delta$, but the primes congruent to 1 mod 4 and 3 mod 4 respectively make up density $1/2$ subsets of primes. You can also get secondary, tertiary, or as many error terms as you wish.