Core partitions have attracted much attention since Anderson's work (2002) on the number of $(s,t)$-core partitions for coprime $s,t$. Recently, there has been a growing interest in studying the limiting distributions of the sizes of random simultaneous core partitions. In this paper, we prove the asymptotic normality of certain statistics of uniform random core partitions with bounded perimeters in the Kolmogorov and Wasserstein $W_1$ distances, including the length and size of a random (strict) $n$-core partition, the length of the Durfee square and the size of a random self-conjugate $n$-core partition. Accordingly, we prove that these statistics are subgaussian. This contrasts with the asymptotic behavior of the size of a random $(s, t)$-core partition for coprime $s,t$ studied by Even-Zohar (2022), which converges in law to Watson's $U^2$ distribution. Our results show that the distribution of the size of a random strict $(n, dn+1)$-core partition is asymptotically normal when $d \ge 3$ is fixed and $n$ tends to infinity, which is an analog of Zaleski's conjecture (2017) and covers Komlós, Sergel, and Tusnády's result (2020) as a special case. Our proof integrates a variety of combinatorial and probabilistic tools, including Stein's method based on Hoeffding decomposition, Hoeffding's combinatorial central limit theorem, the Efron-Stein inequalities on product spaces and slices, and asymptotics of Pólya frequency sequences. Furthermore, our approach is potentially applicable to the study of the asymptotic normality of functionals of random variables with certain global dependence structures that can be decomposed into appropriate mixture forms.