Let $C$ be a locally $\kappa$-presentable category. Then we may write $C = |C_\bullet|$ where each $C_n$ is a presheaf category and the geometric realization is taken in $Pr^L_\kappa$ (and so also is a geometric realization in $Pr^L$). So if $D$ is $\kappa$-presentable, then $LAdj(C,D) = LAdj(|C_\bullet|,D) = Tot LAdj(C_\bullet,D)$, where the totalization is taken in $Pr^L$ (and the cosimplicial diagram lives in $Pr^L_\kappa$ [1]). Now, $Pr^L_\kappa$ is closed in $Pr^L$ under $\kappa$-small limits if $\kappa$ is uncountable, and the simplex category is countable. So if $\kappa$ is uncountable, then $LAdj(C,D)$ is again $\kappa$-presentable.
( 1-categorically, I suppose this might even work if $\kappa = \omega$, since the cosimplicial object can be truncated at a finite stage -- but I'm not sure that $Pr^L_\omega$ is closed under finite limits in $Pr^L$.)

[1] This is actually the subtlest step, I think. By the presheaf case, $LAdj(C_n,D)$ is locally $\kappa$-presentable. The subtle part is verifying that the transition maps $LAdj(C_n,D) \to LAdj(C_m,D)$ preserve $\kappa$-presentable objects. Note that if $A$ is a small category and $D$ is locally $\kappa$-presentable, then the $\kappa$-presentable objects of $D^A$ are those functors $A \to D$ which are left Kan extended from a functor $B \to D_\kappa$ where $B$ is $\kappa$-small and $D_\kappa \subseteq D$ comprises the $\kappa$-presentable objects. The relevant functors $C_{n+1} \to C_n$ with which we are precomposing all have fully faithful right adjoints $\iota$, so if $C_n \to D$ is extended from $B$, then the composite $C_{n+1} \to C_n \to D$ is extended from $\iota(B)$; thus the precomposition functor $LAdj(C_n,D) \to LAdj(C_{n+1},D)$ does indeed preserve $\kappa$-presentable objects.