How to extract a particular element from a json array in postgres?

One approach is to explode your array in fields and to query them. The trick is to consider only the fields of interest.

Select myOutputField 
from json_to_recordset('[the array]') as (myQueryField text, myOutputField text)
where myQueryField = myCondition;

Or, bound to your example:

select "defaultValue" from json_to_recordset('
[
{"paramName":"param1",
 "type":"string",
 "defaultValue":"tomcat7",
 "optional":false,
 "deploymentParam":false},    
{"paramName":"param123PreStopAction",
 "type":"path",
 "defaultValue":"HELLO",
 "optional":false,
 "deploymentParam":false}
]') as x("paramName" text,"defaultValue" text)
where "paramName" =  'param123PreStopAction';

** EDIT ** Your data is not saved in a json column but in a text column. You would have to convert it to json (ideally, the column itself... or at least its content). Also, the json_to_recordset works on single items, not on sets, so you would need to use a LATERAL JOIN to overcome this limitation, as nicely explained here.

SELECT myOutputField 
FROM mytable a
     CROSS JOIN LATERAL
      json_to_recordset(a.jsonintextcolumn::json) as (myQueryField text, myOutputField text)
WHERE myQueryField = myCondition;

Or, bound to your example:

SELECT "defaultValue" 
FROM public.testjsontxt a
     CROSS JOIN LATERAL
     json_to_recordset(a.param_specs::json) as x("paramName" text,"defaultValue" text)
WHERE "paramName" =  'param123PreStopAction';