How do I use the priority list of substituent groups to decide the parent chain for this compound?
The most important simplified criteria for the choice of a principal chain are:

greater number of substituents corresponding to the suffix (principal characteristic group)
longest chain
greater number of multiple bonds
lower locants for suffixes
lower locants for multiple bonds
greater number of prefixes
lower locants for prefixes
lower locants for substituents cited first as a prefix in the name

The corresponding wording of the rules taken from Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) is as follows.

P-44.1 SENIORITY ORDER FOR PARENT STRUCTURES
When there is a choice, the senior parent structure is chosen by applying the following criteria, in order, until a decision is reached. These criteria must always be applied before those applicable to rings and ring systems (see P-44.2) and to chains (see P-44.3). Then criteria applicable to both chains and rings or ring systems given in P-44.4 are considered.
P-44.1.1 The senior parent structure has the maximum number of substituents corresponding to the principal characteristic group (suffix) or senior parent hydride in accord with the seniority of classes (P-41) and the seniority of suffixes (P-43).
(…)
P-44.3.2 The principal chain has the greater number of skeletal atoms [criterion (b) in P-44.3].
(…)
P-44.4.1 If the criteria of P-44.1 through P-44.3, where applicable, do not effect a choice of a senior parent structure, the following criteria are applied successively until there are no alternatives remaining. These criteria are illustrated in P-44.4.1.1 through P-44.4.1.12.
The senior ring, ring system, or principal chain:
(a) has the greater number of multiple bonds (P-44.4.1.1);
(b) has the greater number of double bonds (P-44.4.1.2);
(…)
(h) has the lower locant for an attached group expressed as a suffix (P-44.4.1.8);
(…)
(j) has the lower locant(s) for endings or prefixes that express changes in the level of hydrogenation, i.e., for ‘ene’ and ‘yne’ endings and ‘hydro/dehydro’ prefixes (P-44.4.1.10);
(…)
P-45.2.1 The preferred IUPAC name is based on the senior parent structure that has the maximum number of substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
P-45.2.2 The preferred IUPAC name is based on the senior parent structure that has the lower locant or set of locants for substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
P-45.2.3 The preferred IUPAC name is based on the senior parent structure that has the lower locant or set of locants for substituents cited as prefixes to the parent structure (other than ‘hydro/dehydro’ prefixes) in their order of citation in the name.
(…)

So, yes, the priority list of substituent groups has to be considered first in order to decide the parent chain for this compound according to Rule P-44.1.1. However, you have already done that in both of your suggested examples: The principal characteristic group that is expressed as a suffix is the acid group ($\ce{-COOH}$). Therefore, the parent structure is a butanoic acid.
Looking at your two examples, your actual question is about which of the two possible butanoic acids is the principal chain: 3-chloro-2-(2-hydroxyethyl)butanoic acid or 2-(1-chloroethyl)-4-hydroxybutanoic acid. Thus, your actual question is about how to identify the principal chain after you have already considered the priority order of functional groups.
Both possible chains have the same length; therefore, the next rule (P-44.3.2) cannot be used to make a decision.
Both possible chains don’t have any multiple bonds; therefore, also Rule P-44.4.1 (a and b) doesn’t help.
In both possible chains, the principal characteristic group ($\ce{-COOH}$) is located at the start of the chain; thus, also Rule P-44.4.1 (h) doesn’t make any difference.
The number of substituents expressed as prefixes is the same (two) in both cases: chloro and 2-hydroxyethyl compared to chloroethyl and hydroxy; therefore, Rule P-45.2.1 cannot decide which chain is the principal one.
The decision is finally made using Rule P-45.2.2 since the locants et ‘2,3’ in 3-chloro-2-(2-hydroxyethyl)butanoic acid is lower than ‘2,4’ in 2-(1-chloroethyl)-4-hydroxybutanoic acid.
Therefore, the preferred IUPAC name for the compound that is given in the question is 3-chloro-2-(2-hydroxyethyl)butanoic acid.