Complex MySQL query through relational tables
Looks like straightforward addition of INNER JOIN for every condition will do

SELECT id 
FROM profile p 

INNER JOIN profile_attribute_values pav1 ON p.id = pav1.profile_id
  AND pav1.attribute_id = '58' 
INNER JOIN profiles p1 ON p1.id = pav1.parent_profile_id 
  AND p1.`name` = 'Douglas-Connelly'

INNER JOIN profile_attribute_values pav2 ON p.id = pav2.profile_id
  AND pav2.attribute_id = '26' 
INNER JOIN profiles p2 ON p2.id = pav2.parent_profile_id 
  AND p2.`name` = 'Brekke LLC'

INNER JOIN profile_attribute_values pav3 ON p.id = pav3.profile_id
  AND pav3.ne_attribute_id = '35' AND pav3.val = 'ASA'

INNER JOIN profile_attribute_values pav4 ON p.id = pav4.profile_id
  AND pav4.ne_attribute_id = '38' AND pav4.val = 'Stanton'