Sasho basically answered everything, but I'll add a bit.

Counting the number of lattice points in a symmetric convex body is strictly harder than telling whether there exists a non-zero lattice point inside a convex body, which is the Shortest Vector Problem and is NP-hard (under randomized reductions), even when the body is a Euclidean ball (or any $\ell_p$ ball), and even for any constant-factor approximation [1].

In fact, SVP is widely believed to be hard (but not NP-hard) to approximate even to within any polynomial approximation factor, and a lot of cryptography is based on this assumption. (Some cryptography is based on the presumed hardness for even superpolynomial approximation factors.) The best-known polynomial-time approximation factor is $2^{C n \log \log n/\log n}$ for any constant $C > 0$, which follows from a long line of work. (This algorithm is in the Euclidean norm, so to extend even this to arbitrary norms, you must be able to calculate a not-too-terrible approximating ellipsoid.) It would be a major breakthrough to improve this.

Finding the parity of the number of lattice points in a shifted Euclidean ball (or any $\ell_p$ ball) is in fact as hard as finding the parity of the number of solutions to a $2$-SAT instance. (For centrally symmetric bodies, the parity is always odd, so the problem is only interesting for shifted bodies or asymmetric bodies.) This follows from the simple reduction from Max-2-SAT in [2] with Bennett and Golovnev, which preserves the number of solutions, as we mention at the end of Section 6. There is probably an earlier reduction that also has this property.

Finally, there are many of upper bounds on the number of lattice points in a convex body based on certain geometric parameters of the body. For example, Henk's bound [3]. The one that Sasho described is probably the easiest to work with.

[1] *Khot, Subhash*, **Hardness of approximating the shortest vector problem in lattices**, J. ACM 52, No. 5, 789-808 (2005). ZBL1323.68301.

[2] https://arxiv.org/abs/1704.03928

[3] *Henk, Martin*, **Successive minima and lattice points**, Schneider, Rolf (ed.) et al., IV international conference on “Stochastic geometry, convex bodies, empirical measures and applications to engineering science”, Tropea, Italy, September 24--29, 2001. Vol. I. Palermo: Circolo Matematico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 70, 377-384 (2002). ZBL1126.11034.