What is the senior parent chain in the following compound?
The most important simplified criteria for the choice of a principal chain are:


greater number of suffixes
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.
  
  (…)


You have correctly identified the suffix (“oic acid”) for the principal characteristic group. The next criterion for the principal chain is the greater number of skeletal atoms (i.e. the longest chain). Thus, the principal chain corresponds to the but-3-ynoic acid part and not to the 3-hydroxypropanoic acid part since but-3-ynoic acid has a longer chain than propanoic acid. Therefore, the correct name is 2-(hydroxymethyl)but-3-ynoic acid.