See if changing to y = DataInterpolations.CubicSpline(pars, eltype(pars).(extra[1])).(extra[2]) fixes the ForwardDiff one. It might be missing a type promotion within the spline construction and thus require the two elements are the same type (which would be easy to fix).
As for the Zygote one, try running the following before using any Zygote in a fresh session:
ZygoteRules.@adjoint ZygoteRules.literal_getproperty(A::Tridiagonal, ::Val{:dl}) = A.dl, y -> Tridiagonal(dl, zeros(length(d)), zeros(length(du)),)
ZygoteRules.@adjoint ZygoteRules.literal_getproperty(A::Tridiagonal, ::Val{:d}) = A.d, y -> Tridiagonal(zeros(length(dl)), d, zeros(length(du)),)
ZygoteRules.@adjoint ZygoteRules.literal_getproperty(A::Tridiagonal, ::Val{:du}) = A.dl, y -> Tridiagonal(zeros(length(dl)), zeros(length(d), du),)
ZygoteRules.@adjoint Tridiagonal(dl, d, du) = Tridiagonal(dl, d, du), p̄ -> (diag(p̄[2:end, 1:end-1]), diag(p̄), diag(p̄[1:end-1, 2:end]))