One way that you can do is to download the template file which contains parameter file too using Export-AzResourceGroup. Below is the script that worked for me.
$resource = Get-AzResource -ResourceGroupName "<YOUR_RESOUREGROUP_NAME>" -ResourceType 'Microsoft.Web/sites' -Name "<YOUR_LOGICAPP_NAME>"
$resource.ResourceId
Export-AzResourceGroup -ResourceGroupName '<YOUR_RESOUREGROUP_NAME>' -Resource $resource.ResourceId -IncludeParameterDefaultValue -Force
However, if you want to retrieve only parameter file, you can use the below script that worked fine when reproduced from my end.
$resource = Get-AzResource -ResourceGroupName "<YOUR_RESOUREGROUP_NAME>" -ResourceType 'Microsoft.Web/sites' -Name "<YOUR_LOGICAPP_NAME>"
$resource.ResourceId
Export-AzResourceGroup -ResourceGroupName '<YOUR_RESOUREGROUP_NAME>' -Resource $resource.ResourceId -IncludeParameterDefaultValue -Force
$fileContent = Get-Content -Path '<PATH_TO_EXPORTED_FILE>' | ConvertFrom-Json
$schema = '"'+"`$schema"+'"'+":"+'"'+$fileContent.'$schema'+'"'
$contentVersion = '"'+"contentVersion"+'"'+":"+'"'+$fileContent.contentVersion+'"'
$parameter = $fileContent.parameters | ConvertTo-Json
$parameters = '"'+"parameters"+'"'+":"+$parameter
$parameterFile = $schema+","+$contentVersion+","+$parameters
$parameterFileJson = "{"+$parameterFile+"}" | ConvertFrom-Json | ConvertTo-Json
$parameterFileJson | Out-File '<PATH>/parameters.json'
Results:
