How to identify the parent chain if two chains have the same number of skeletal atoms and the same number of double bonds?
The most important simplified criteria for the choice of a principal chain in such compounds are:


greater number of suffixes
longest chain
greater number of multiple bonds
lower locants for suffixes
lower locants for multiple bonds


The corresponding wording of the nomenclature rules (taken from Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book)) reads 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);
  
  (…)


Since the compound that is given in the question does not have any principal characteristic group (a characteristic group chosen for citation at the end of a name by means of a suffix or a class name, or implied by a trivial name), the first criterion (greater number of suffixes) given in P-44.1.1 is not relevant in this case.

The second criterion given in P-44.3.2 is the longest chain. However, the  chains A–C and B–C both have eight skeletal atoms.

Also, both chains have three double bonds. Therefore, the criteria given in P-44.4.1 (a) (greater number of multiple bonds) and (b) (greater number of double bonds) cannot be used to reach a decision.

The criterion given in P-44.4.1 (h) (lower locant for the suffix) is not relevant since the compound that is given in the question does not have any suffix.

Finally, the criterion given in P-44.4.1 (j) stipulates that the principal chain shall have the lower locants for the double bonds. Therefore, the principal chain for the compound that is given in the question is the octa-1,4,6-triene from A to C rather than the octa-2,4,6-triene from B to C since the locant set ‘1,4,6’ is lower than ‘2,4,6’.

Thus, the complete name for the compound that is given in the question is (3⁠E,4⁠E,6⁠E)-3-ethylideneocta-1,4,6-triene.