It should also be storing address of GetProcAddress() at <lpparm+4>

it should store a string the name of module at <lpparam+14>

it should store a string the name of proc for getProcAddress at <lpparm+46>

at [esp+8] it accesses the lpparam which was written to the remote process so you may need to find this WriteProcessMemory Also and look at the local Buffer to Know the data ie name of dll and name of Proc

or you may need to attach to the Remote process and Break on Thread CreateEvent and single Step through the BaseThreadStartThunk() Routine in this routine you will notice ebp is Zeored in there and Eax and Ebx holds LpStart and lpParam and that is how lpStart gets Executed

push [ebp+c] lpparam
call [enp+8] lpStart

esi gets the lpparm via mov esi [esp+8] which in your case holds the Address of LoadLibrary as you correctly observed but you have not observed what is at

assuming [esp+8 ] holds 403000 <lpparam>
[[esp+8] + 0 ] [403000] holds Address of LoadLibrary as you observed
[[esp+8] + 4]  [403004] will hold Address of GetProcAddress
[[esp+8] + 14] [403014] will hold a string the dll name
[[esp+8] + 46] [403046] will hold the name of  Proc in the Loaded Dll
[[esp+8] + 10] [403010] gets the Result of LoadLibrary Return in your above snippet

It should also be storing address of GetProcAddress() at <lpparm+4>

it should store a string the name of module at <lpparam+14>

it should store a string the name of proc for getProcAddress at <lpparm+46>

at [esp+8] it accesses the lpparam which was written to the remote process so you may need to find this WriteProcessMemory Also and look at the local Buffer to Know the data ie name of dll and name of Proc

or you may need to attach to the Remote process and Break on Thread CreateEvent and single Step through the BaseThreadStartThunk() Routine in this routine you will notice ebp is Zeored in there and Eax and Ebx holds LpStart and lpParam and that is how lpStart gets Executed

push [ebp+c] lpparam
call [enp+8] lpStart

esi gets the lpparm via mov esi [esp+8] which in your case holds the Address of LoadLibrary as you correctly observed but you have not observed what is at

assuming [esp+8 ] holds 403000 <lpparam>
[[esp+8] + 0 ] [403000] holds Address of LoadLibrary as you observed
[[esp+8] + 4]  [403004] will hold Address of GetProcAddress
[[esp+8] + 14] [403014] will hold a string the dll name
[[esp+8] + 46] [403046] will hold the name of  Proc in the Loaded Dll
[[esp+8] + 10] [403010] gets the Result of LoadLibrary Return in your above snippet

edit

added the annotated snippet for referance

push esi
 mov esi, [esp+8]    <lpparameter> assuming 403000
 lea eax, [esi+14h]  lpparameter+14> should be a string the name of dll 403014
 push eax            push 403014    
 call dword ptr[esi] DWORD [ lpparmeter+0 ] = [403000] = Address Of LoadLib so calling loadlib
 test eax, eax
 mov [esi+10h], eax  [lpparam+10] [403010] store LoadLibResult here
 jz short loc_402B80  failure
 lea ecx, [esi+46h] [403046] = <lpparm+46> again a string name of Proc for GetProcAddress()
 push ecx           processname     
 push eax           HandletoLoadedmod  
 call dword ptr [esi+4]  Must be Address of GetProcAddress from the looks check your code 
capture local buffer when Writing to ProcessMemory or Break on Create new Thread in the RemoteProcess
by attaching a debugger Executing the RemoteDebugge and when a new Thread Create Event is triggered 
single step to find the contents of lpparam
 test eax, eax    
 jz short_loc402B80
 call eax  <calls the  functions that was just getproc'ed>
 push 0
 call