<table-valued function> is not a recognized built-in function name

It's a table-valued function. So you probably meant:

SELECT p.DOCTORFISTNAME, p.DOCTORLASTNAME, t.FirstName, t.Middle, t.LastName
  FROM dbo.[PracticeandPhysician] AS p
  CROSS APPLY dbo.fnParseName(p.DOCTORFIRSTNAME + ' ' + p.DOCTORLASTNAME);

Note that you can't say:

SELECT dbo.TableValueFunction('foo');

Any more than you could say:

SELECT dbo.Table;
--or
SELECT dbo.View;

You can, however, say:

SELECT * FROM dbo.fnParseName('foo bar');
--or
SELECT FirstName, Middle, LastName FROM dbo.fnParseName('foo bar');

(Not that I have validated that your function does what you think, or does so efficiently.)

Please always use the dbo. prefix as others have suggested.

UDFs/Functions need to be prefixed with the schema name (most likely "dbo"). Change the call to

SELECT
    dbo.fnParseName(DOCTORFIRSTNAME + ' ' + DOCTORLASTNAME) 
FROM
    [PracticeandPhysician]

The problem you have is similar to what I encountered too. Scalar function and Table inline functions are quite different in terms of implementation. See below for the diiferent

 Create function udfCountry
 (
@CountryName varchar(50)
 )
 returns varchar(2)
 as 
 BEGIN
 Declare @CountryID varchar(2),
    @Result varchar(2)

 Select @CountryID = Country from 
dbo.GeoIPCountryNames where CountryName = @CountryName

set @Result = isNull(@CountryID, 'NA')
if @Result = 'NA'
set @Result = 'SD'
return @Result
End

//Implementation

select dbo.[udfCountry]('Nigeria')

// sample result

NG

// Inline table function sample

 Create FUNCTION  ConditionEvaluation
 (
  @CountrySearch varchar(50)
 )
 returns @CountryTable table 
(
Country varchar(2),
CountryName varchar(50)
 )
 as 

 Begin

 Insert into @CountryTable(Country, CountryName)
 Select Country, CountryName from GeoIPCountryNames
 where Country like '%'+@CountrySearch+'%'
 return 
 end

//Implementation sample

 Declare @CountrySearch varchar(50)
 set @CountrySearch='a'
 select * from  ConditionEvaluation(@CountrySearch)

the parttern of implementating scalar is quite different inline table. I hope this helps

You always have to prefix SQL function calls with the schema name dbo. or the schema name for that function (dbo is the default schema).

SELECT dbo.fnParseName(--etc