Your WSDL states that it has to be called in this way:
$this->client->SearchFlights(array('searchRequest' => $parameters));
Note that the searchRequest name is important.
There are also various mandatory inputs for that service, you should ensure that those are present. From looking at your code, you are missing some.
Take a look at the FlightSearchRequest complex type here. Each param that has minOccurs=1 is required.
<s:complexType name="FlightSearchRequest">
<s:sequence>
<s:element minOccurs="1" maxOccurs="1" name="TypeOfTrip" type="tns:TripType"/>
<s:element minOccurs="0" maxOccurs="1" name="FromCity" type="s:string"/>
<s:element minOccurs="0" maxOccurs="1" name="ToCity" type="s:string"/>
<s:element minOccurs="0" maxOccurs="1" name="ReturnFromCity" type="s:string"/>
<s:element minOccurs="0" maxOccurs="1" name="ReturnToCity" type="s:string"/>
<s:element minOccurs="1" maxOccurs="1" name="DepartureDate" type="s:dateTime"/>
<s:element minOccurs="1" maxOccurs="1" name="ReturnDate" type="s:dateTime"/>
<s:element minOccurs="1" maxOccurs="1" name="NumberOfAdults" type="s:int"/>
<s:element minOccurs="1" maxOccurs="1" name="NumberOfChildren" type="s:int"/>
<s:element minOccurs="1" maxOccurs="1" name="NumberOfInfantsInLap" type="s:int"/>
<s:element minOccurs="1" maxOccurs="1" name="NumberOfInfantsOnSeat" type="s:int"/>
<s:element minOccurs="1" maxOccurs="1" name="NumberOfSeniors" type="s:int"/>
<s:element minOccurs="0" maxOccurs="1" name="AirlinePreference" type="s:string"/>
<s:element minOccurs="1" maxOccurs="1" name="ClassOfService" type="tns:FlightClass"/>
<s:element minOccurs="1" maxOccurs="1" name="IsNonStop" type="s:boolean"/>
<s:element minOccurs="1" maxOccurs="1" name="ConsolidatorFaresOnly" type="s:boolean"/>
<s:element minOccurs="0" maxOccurs="1" name="FpAffiliate" type="s:string"/>
<s:element minOccurs="0" maxOccurs="1" name="FpSubAffiliate" type="s:string"/>
</s:sequence>
</s:complexType>