Configure Connection String to Use Target Machine Credentials
The dtsx package runs where you install and execute it. Usually in SSISDB.
Then you can create a sql server agent job to run it.
The connection string specifies where the action (executing some sql script, or as a data source/destination in a dataflow as examples) will be executed.
You can install the package in every server you want and then refer in the connection string for example to localhost to run it against the same server.
You can also manage parameters and design a parametric connection string as you pointed out.
If you have a list of servers to contact with the connection string, you have to loop into the list and place the values in the connection string and then execute the task you want using parameterized connection string. So you will contact every server you want and execute what you need.
But there will be always only one server hat runs the solution.
If you have a job in every of your servers that point to a local SSIDB with a local dtsx to execute against a local instance you have to manage them with a local environment that define a good connection string for each one of them.