Nomenclature question about diols
I'll be using the brief guide to the nomenclature of Organic Chemistry as a reference, because I could not gain access to the blue book. This source suffices for this nomenclature example.

Seniority of parent compounds
The systematic name is based on the name of the senior parent compound, which is chosen by applying the following criteria in the order described below and shown in Fig. 1, until a decision is reached. For a complete set of criteria see Ref. [8].


In the given example, both the chains contain the senior most group, which is the hydroxyl (alcohol) group. The smaller chain, however, contains 2 hydroxyl groups as compared to the larger chain, which contains one hydroxyl and one alkylhydroxyl group (which is a lower priority). Since this is the first point of difference, The smaller chain would be the parent compound.
After identifying the parent chain, it is easy to name the locants such that the seniormost groups get lower locants; As you have written, the name would finally be 2-ethylhexane-1,3-diol.
This image is possibly more detailed than this question requires, but the image and the linked reference are good resources, should you need more information on naming tricky compounds.